9117:
5867:
90:
4201:
7549:
11613:
31:
1360:
10203:
12192:
7255:
5619:
7883:
12162:
2792:
4825:
11305:
10149:
9067:
4882:
1800:
8839:
1234:
11911:
6497:
9848:
9412:
5267:
11943:
7544:{\displaystyle \Phi (R)=\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{0}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} +\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{C}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} }
2540:
4576:
5862:{\displaystyle \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\rho (\mathbf {r} '){\frac {\mathbf {r} -\mathbf {r} '}{\left\|\mathbf {r} -\mathbf {r} '\right\|^{3}}}\mathrm {d} \mathbf {r} '\equiv {\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}
8149:
7655:
7182:
5488:
1579:
10040:
3650:
7040:
11608:{\displaystyle {\frac {\left({\cfrac {q^{2}}{4\pi \varepsilon _{0}L_{1}^{2}}}\right)}{\left({\cfrac {\frac {q^{2}}{4}}{4\pi \varepsilon _{0}L_{2}^{2}}}\right)}}={\frac {mg\tan \theta _{1}}{mg\tan \theta _{2}}}\Rightarrow 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}={\frac {\tan \theta _{1}}{\tan \theta _{2}}}}
8646:
11752:
6600:
3705:
because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. Charge is always discrete in reality, and the "continuous charge" assumption is just an approximation
9038:, due to the validity of Coulomb's law within its specific range of application. Also note that the spherical symmetry for gauss law on stationary charges is not valid for moving charges owing to the breaking of symmetry by the specification of direction of velocity in the problem. Agreement with
6351:
11119:
5119:
4139:
9214:
9590:
8642:
3937:
5575:
Since
Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and, in this respect, Gauss's law is more general than Coulomb's law.
7980:
1120:
6270:
11743:
is sufficient to verify that the equality is true taking into account the experimental error. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently great, the angles will be small enough to make the following approximation:
1969:
5376:
10614:
12157:{\displaystyle {\frac {\frac {L_{1}}{2\ell }}{\frac {L_{2}}{2\ell }}}\approx 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}\Rightarrow {\frac {L_{1}}{L_{2}}}\approx 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}\Rightarrow {\frac {L_{1}}{L_{2}}}\approx {\sqrt{4}}}
8458:
5360:
2787:{\displaystyle \mathbf {F} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={q \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}},}
9000:(yet without violating relativistic-energy momentum conservation). Note that the expression for electric field reduces to Coulomb's law for non-relativistic speeds of the point charge and that the magnetic field in non-relativistic limit (approximating
5043:
4820:{\displaystyle \mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}}}
8355:
9891:
5569:
7878:{\displaystyle \Phi (R)={\frac {Q(R)}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}\rho (\mathbf {r} '){\mathrm {d} \mathbf {r} '}={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} '_{c})|B_{R}(\mathbf {r} _{0})|}
3516:
7045:
7645:
11283:
9537:
10977:
1795:{\displaystyle \mathbf {F} _{1}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{12} \over {|\mathbf {r} _{12}|}^{2}}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{\mathbf {r} _{12} \over {|\mathbf {r} _{12}|}^{3}}}
8267:
4004:
6902:
1174:, the Greek word for "amber") to refer to the property of attracting small objects after being rubbed. This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in
10880:
1087:
The force is along the straight line joining the two charges. If the charges have the same sign, the electrostatic force between them makes them repel; if they have different signs, the force between them makes them attract.
9209:
8505:
6665:
6006:
8834:{\displaystyle \mathbf {B} ={\frac {q}{4\pi \epsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}}
3810:
8489:
which, unlike
Coulomb's law is not limited to stationary test charges. Considering the charge to be invariant of observer, the electric and magnetic fields of a uniformly moving point charge can hence be derived by the
6344:
6175:
1103:, but gravitational forces always make things attract, while electrostatic forces make charges attract or repel. Also, gravitational forces are much weaker than electrostatic forces. Coulomb's law can be used to derive
11906:{\displaystyle \tan \theta \approx \sin \theta ={\frac {\frac {L}{2}}{\ell }}={\frac {L}{2\ell }}\Rightarrow {\frac {\tan \theta _{1}}{\tan \theta _{2}}}\approx {\frac {\frac {L_{1}}{2\ell }}{\frac {L_{2}}{2\ell }}}}
6502:
7248:
947:
It follows therefore from these three tests, that the repulsive force that the two balls â electrified with the same kind of electricity â exert on each other, follows the inverse proportion of the square of the
942:
is directly proportional to the product of the magnitudes of their charges and inversely proportional to the square of the distance between them. Coulomb discovered that bodies with like electrical charges repel:
10770:
3147:
6182:
3467:
6492:{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla _{\mathbf {r} }\cdot e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}=0}
4486:
3309:
1049:
9539:
where we look at the (connected) S-matrix entry for two electrons scattering off each other, treating one with "fixed" momentum as the source of the potential, and the other scattering off that potential.
66:
is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like charges repel each other, and opposite charges attract each
10393:
10499:
4571:
5262:{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {s} )(\mathbf {r} -\mathbf {s} )}{|\mathbf {r} -\mathbf {s} |^{3}}}\,\mathrm {d} ^{3}\mathbf {s} }
8376:
7947:
6757:
12398:
Il rĂ©sulte donc de ces trois essais, que l'action rĂ©pulsive que les deux balles Ă©lectrifĂ©es de la mĂȘme nature d'Ă©lectricitĂ© exercent l'une sur l'autre, suit la raison inverse du carrĂ© des distances.
1347:, and a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the
9407:{\displaystyle {\mathcal {A}}(|\mathbf {p} \rangle \to |\mathbf {p} '\rangle )-1=2\pi \delta (E_{p}-E_{p'})(-i)\int d^{3}\mathbf {r} \,V(\mathbf {r} )e^{-i(\mathbf {p} -\mathbf {p} ')\mathbf {r} }}
5278:
4961:
9843:{\displaystyle \langle p',k|S|p,k\rangle |_{conn}=-i{\frac {e^{2}}{|\mathbf {p} -\mathbf {p} '|^{2}-i\varepsilon }}(2m)^{2}\delta (E_{p,k}-E_{p',k})(2\pi )^{4}\delta (\mathbf {p} -\mathbf {p} ')}
8274:
11189:
5941:
3755:
3703:
3805:
10967:
1866:
10461:
5499:
2205:
8144:{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} _{0})=\lim _{R\to 0}{\frac {1}{|B_{R}(\mathbf {r} _{0})|}}\Phi (R)={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} _{0})}
13531:
10123:
3027:
1351:. By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls and derive his inverse-square proportionality law.
6706:
4947:. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space).
4325:
3189:
2318:
1857:
5903:
3347:
6103:
4394:
can be derived from
Coulomb's law. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the
2424:
9586:
5614:
4170:
force. For slow movement, the magnetic force is minimal and
Coulomb's law can still be considered approximately correct. A more accurate approximation in this case is, however, the
2916:
1261:
of
England. In his notes, Cavendish wrote, "We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the
12842:
8481:. When no acceleration is involved in a particle's history, Coulomb's law can be assumed on any test particle in its own inertial frame, supported by symmetry arguments in solving
2097:
11194:
6865:
3374:
3216:
3054:
10066:
3511:
11741:
11712:
10688:
10659:
6894:
6786:
6042:
5483:{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\int \rho (\mathbf {s} )\,\delta (\mathbf {r} -\mathbf {s} )\,\mathrm {d} ^{3}\mathbf {s} }
3999:
3964:
2945:
2057:
1574:
1518:
10163:
13171:
9417:
8209:
2856:
2126:
1462:
10335:
10313:
8987:
8965:
8903:
8881:
7177:{\displaystyle \mathbf {E} _{C}={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega \setminus B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}
3082:
2878:
2513:
11683:
11656:
10923:
9024:
6071:
2355:
2253:
10796:
4285:
4236:
2471:
9886:
8187:
is spherically symmetric (this assumption, like
Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).
7556:
3402:
3244:
8943:
6826:
13359:
Spavieri, G., Gillies, G. T., & Rodriguez, M. (2004). Physical implications of
Coulombâs Law. Metrologia, 41(5), S159âS170. doi:10.1088/0026-1394/41/5/s06
8923:
8498:
on the test charge in the charge's frame of reference given by
Coulomb's law and attributing magnetic and electric fields by their definitions given by the form of
7973:
4519:
2821:
2382:
2280:
2153:
1545:
1489:
10035:{\displaystyle \int V(\mathbf {r} )e^{-i(\mathbf {p} -\mathbf {p} ')\mathbf {r} }d^{3}\mathbf {r} ={\frac {e^{2}}{|\mathbf {p} -\mathbf {p} '|^{2}-i\varepsilon }}}
1107:, and vice versa. In the case of a single point charge at rest, the two laws are equivalent, expressing the same physical law in different ways. The law has been
10291:
4383:
4263:
2988:
2028:
2000:
10268:
10248:
10228:
8859:
6806:
3489:
2491:
647:
7035:{\displaystyle \mathbf {E} _{B_{R}}={\frac {1}{4\pi \varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}
2440:
allows
Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the
4356:
2533:
8473:
Coulomb's law can be used to gain insight into the form of the magnetic field generated by moving charges since by special relativity, in certain cases the
3645:{\displaystyle \mathbf {F} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}\int dq'{\frac {\mathbf {r} -\mathbf {r'} }{|\mathbf {r} -\mathbf {r'} |^{3}}}.}
1230:. However, he did not generalize or elaborate on this. In 1767, he conjectured that the force between charges varied as the inverse square of the distance.
10698:
4166:. When movement takes place, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the
12180:
In this way, the verification is limited to measuring the distance between the charges and checking that the division approximates the theoretical value.
620:
4204:
If two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.
1257:
In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by
4417:
4358:, the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge
955:
632:
12949:
10345:
9085:
13245:
13092:
6595:{\displaystyle \Omega \cap V=\emptyset \implies \forall \mathbf {r} \in V\ \ \forall \mathbf {r'} \in \Omega \ \ \ \mathbf {r} \neq \mathbf {r'} }
11114:{\displaystyle F_{2}={\frac {{({\frac {q}{2}})}^{2}}{4\pi \varepsilon _{0}L_{2}^{2}}}={\frac {\frac {q^{2}}{4}}{4\pi \varepsilon _{0}L_{2}^{2}}}}
9150:
6605:
5946:
1308:
published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the
13349:
13302:
13220:
12899:
12712:
12438:
4134:{\displaystyle k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}=8.987\,551\,792\,3\,(14)\times 10^{9}\ \mathrm {N{\cdot }m^{2}{\cdot }C^{-2}} .}
4932:
Strictly speaking, Gauss's law cannot be derived from
Coulomb's law alone, since Coulomb's law gives the electric field due to an individual,
6277:
6108:
7187:
12813:
3087:
13515:
12834:
12242:
10167:
8637:{\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}\mathbf {r} }
8161:
Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the
3407:
2444:
of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the
3932:{\displaystyle \mathbf {F} _{1}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{12} \over {|\mathbf {r} _{12}|}^{2}}}
883:
652:
13064:
3249:
10419:
12236:
9137:
1227:
1100:
662:
10071:
1248:
announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as
10128:
However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.
4213:
4524:
4925:
487:
8485:, shown above. Coulomb's law can be expanded to moving test particles to be of the same form. This assumption is supported by
6711:
1324:
is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
13485:
13458:
13421:
13325:
13278:
13204:
13165:
13058:
12979:
12933:
12876:
12706:
12662:
12547:
12522:
12497:
12279:
10898:
If we now discharge one of the spheres, and we put it in contact with the charged sphere, each one of them acquires a charge
5273:
is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem
502:
497:
124:
8499:
6265:{\displaystyle \oint _{\partial V}\mathbf {E} _{0}\cdot d\mathbf {S} =\int _{V}\mathbf {\nabla } \cdot \mathbf {E} _{0}\,dV}
12903:
8271:
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
512:
1964:{\textstyle {\hat {\mathbf {r} }}_{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{12}}{|\mathbf {r} _{12}|}}}
11140:
7888:
4868:
is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.
4860:
and for the forces that bind atoms and molecules together to form solids and liquids. Generally, as the distance between
3768:
931:
and maybe even its starting point, as it allowed meaningful discussions of the amount of electric charge in a particle.
13561:
12637:
10928:
13122:
12789:
12469:
12422:
12315:
10189:
9103:
114:
10609:{\displaystyle {\frac {\sin \theta _{1}}{\cos \theta _{1}}}={\frac {F_{1}}{mg}}\Rightarrow F_{1}=mg\tan \theta _{1}}
9035:
5911:
3709:
3657:
2163:
382:
10171:
8453:{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {\hat {\mathbf {r} }}{r^{2}}}}
6676:
297:
4290:
2285:
5355:{\displaystyle \nabla \cdot \left({\frac {\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )}
4331:. More generally, the field can be generated by a distribution of charges who contribute to the overall by the
1807:
1327:
The torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weak
876:
642:
119:
6076:
5038:{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {e} _{r}}{r^{2}}}}
4152:
The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
9546:
8350:{\displaystyle 4\pi r^{2}{\hat {\mathbf {r} }}\cdot \mathbf {E} (\mathbf {r} )={\frac {Q}{\varepsilon _{0}}}}
8176:
4897:
2996:
2388:
657:
362:
6670:
Therefore the flux through a closed surface generated by some charge density outside (the surface) is null.
5586:
3158:
4163:
2883:
1153:
522:
262:
129:
12751:
6834:
5872:
4385:
would move if placed in the field. For a negative point source charge, the direction is radially inwards.
3316:
2066:
252:
12563:
Chalmers, Gordon (1937). "The Lodestone and the Understanding of Matter in Seventeenth Century England".
1305:
1123:
924:
815:
690:
587:
562:
482:
12734:
12679:
5564:{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon _{0}}},}
3654:
The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which
13234:
13088:
Electromagnetic waves, the propagation of potential, and the electromagnetic effects of a moving charge
10045:
9888:
as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain:
8993:
5493:
3494:
2156:
1309:
928:
315:
13525:
12599:
11717:
11688:
10664:
10635:
9543:
Using the Feynman rules to compute the S-matrix element, we obtain in the non-relativistic limit with
9136:, as well as discrete bound states, representing the hydrogen atom. It can also be derived within the
6870:
6762:
6014:
3977:
3942:
2921:
2035:
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1494:
13551:
12863:
1108:
869:
830:
357:
347:
287:
282:
222:
17:
13001:
12696:
10210:
It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass
1431:
the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.
13571:
12340:
3352:
3194:
3032:
1245:
934:
The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic
367:
10318:
10296:
8970:
8948:
8886:
8864:
3151:
For a surface charge distribution (a good approximation for charge on a plate in a parallel plate
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2832:
2496:
2102:
1438:
800:
302:
13556:
12226:
11661:
11634:
9003:
8172:
7640:{\displaystyle (\Omega \setminus B_{R}(\mathbf {r} _{0}))\cap B_{R}(\mathbf {r} _{0})=\emptyset }
4944:
4907:
4332:
2437:
1332:
680:
207:
197:
192:
12809:
10901:
10158:
may contain an excessive amount of intricate detail that may interest only a particular audience
6053:
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Coulomb also showed that oppositely charged bodies attract according to an inverse-square law:
627:
397:
172:
13512:
13396:
13383:
13370:
12629:
12625:
12385:
12362:
9855:
9030:. These solutions, when expressed in retarded time also correspond to the general solution of
4943:
be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the
4268:
4219:
2454:
13538:(video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike.
12216:
11278:{\displaystyle {\frac {\frac {q^{2}}{4}}{4\pi \varepsilon _{0}L_{2}^{2}}}=mg\tan \theta _{2}}
8928:
6811:
4903:
4174:. When the charges are moving more quickly in relation to each other or accelerations occur,
4171:
4148:
There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:
3059:
2216:
1348:
1180:
1137:, could be rubbed with cat's fur to attract light objects like feathers and pieces of paper.
725:
412:
402:
352:
342:
8908:
2963:
over the region containing the charge is equivalent to an infinite summation, treating each
1301:, and there is no reason to think that it differs at all from the inverse duplicate ratio".
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332:
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152:
4856:. This simple law also correctly accounts for the forces that bind atoms together to form
4327:. In the simplest case, the field is considered to be generated solely by a single source
89:
8:
12335:
The development of the concept of electric charge: Electricity from the Greeks to Coulomb
12231:
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9081:
9027:
8183:
be proven from Gauss's law if it is assumed, in addition, that the electric field from a
3967:
3379:
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1860:
923:. Although the law was known earlier, it was first published in 1785 by French physicist
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10213:
9532:{\displaystyle \int {\frac {d^{3}k}{(2\pi )^{3}}}e^{ikr_{0}}\langle p',k|S|p,k\rangle }
9144:
9051:
8997:
8844:
8461:
8262:{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}}
8162:
6791:
4361:
4241:
3474:
2476:
2006:
1978:
1344:
1223:
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12752:"Experiments on Electricity: Experimental determination of the law of electric force."
507:
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The Enlightenment of Joseph Priestley: A Study of his Life and Work from 1733 to 1773
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745:
12871:(9th ed.), International Bureau of Weights and Measures, Dec 2022, p. 15,
4158:
The charges must be stationary with respect to a nonaccelerating frame of reference.
2320:; the charges repel each other. If the charges have opposite signs then the product
13153:
13046:
12967:
12615:
12572:
12457:
10132:
7951:
Where the last equality follows by the mean value theorem for integrals. Using the
4341:
2993:
For a linear charge distribution (a good approximation for charge in a wire) where
2518:
1317:
1219:
1200:
1196:
845:
760:
720:
710:
597:
552:
535:
452:
387:
81:
10875:{\displaystyle {\frac {q^{2}}{4\pi \varepsilon _{0}L_{1}^{2}}}=mg\tan \theta _{1}}
13519:
12779:
12760:(1st ed.). Cambridge, England: Cambridge University Press. pp. 104â113.
12619:
7952:
4891:
4179:
2441:
1313:
1258:
1238:
939:
912:
780:
705:
700:
567:
442:
407:
267:
167:
12891:
10131:
The Coulomb potential, and its derivation, can be seen as a special case of the
8925:
is the ratio of speed of the charged particle divided by the speed of light and
6896:
be the electric field created inside and outside the sphere respectively. Then,
820:
13566:
12247:
12221:
10135:, which is the case where the exchanged boson â the photon â has no rest mass.
8478:
8474:
6048:
4849:
4395:
4195:
2445:
1328:
1104:
904:
740:
735:
557:
447:
372:
322:
272:
245:
202:
177:
147:
140:
13435:
13335:
13288:
13147:
13050:
10661:
be the distance between the charged spheres; the repulsion force between them
4573:
produces an electric field whose magnitude and direction is, by superposition
13545:
13157:
12964:
Discussion on physics teaching innovation: Taking Coulomb's law as an example
12289:
8486:
4865:
2964:
1208:
1175:
855:
840:
825:
765:
477:
392:
377:
292:
277:
182:
13132:
12924:
Serway, Raymond A.; Sewett, John W., Jr. (2014). "Some Physical Constants".
10170:
any relevant information, and removing excessive detail that may be against
9204:{\textstyle {\mathcal {A}}(|\mathbf {p} \rangle \to |\mathbf {p} '\rangle )}
8464:
dependence of the electric field in Coulomb's law follows from Gauss's law.
3313:
For a volume charge distribution (such as charge within a bulk metal) where
1222:
of England was among the first to propose that electrical force followed an
12621:
Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics
10202:
8184:
6660:{\displaystyle \nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}
6001:{\displaystyle \nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}
4955:
4936:
4404:
4328:
4209:
1321:
1096:
835:
730:
695:
637:
572:
492:
457:
337:
212:
46:
4864:
increases, the force of attraction, and binding energy, approach zero and
4212:
that associates to each point in space the Coulomb force experienced by a
1187:
Early investigators of the 18th century who suspected that the electrical
12657:. University Park: Pennsylvania State University Press. pp. 144â56.
8502:. The fields hence found for uniformly moving point charges are given by:
8364:
5058:
1972:
755:
607:
437:
99:
8198:
in the integral form of Gauss's law to be a spherical surface of radius
4155:
The charges must not overlap (e.g. they must be distinct point charges).
1160:
effect from static electricity produced by rubbing amber. He coined the
12971:
12681:
The History and Present State of Electricity, with Original Experiments
10042:
where Fourier transforming both sides, solving the integral and taking
9120:
The most basic Feynman diagram for QED interaction between two fermions
8495:
6339:{\displaystyle e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}
6170:{\displaystyle e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}
2990:. The distribution of charge is usually linear, surface or volumetric.
1111:, and observations have upheld the law on the scale from 10 m to 10 m.
472:
7243:{\displaystyle \mathbf {E} _{B_{R}}+\mathbf {E} _{C}=\mathbf {E} _{0}}
1320:, and determined that the magnitude of the electric force between two
1156:
made a careful study of electricity and magnetism, distinguishing the
13431:
10925:. In the equilibrium state, the distance between the charges will be
10765:{\displaystyle F_{1}={\frac {q^{2}}{4\pi \varepsilon _{0}L_{1}^{2}}}}
8367:
pointing radially away from the charge. Again by spherical symmetry,
6045:
5096:
is the charge of the particle, which is assumed to be located at the
3152:
1204:
1161:
1157:
795:
770:
582:
104:
13117:(3rd ed.). Upper Saddle River, NJ: Prentice Hall. p. 517.
12624:. Los Angeles, California: University of California Press. pp.
8883:
is the position vector from the point source to the point in space,
7650:
The RHS is the electric flux generated by a charged sphere, and so:
915:
particles at rest. This electric force is conventionally called the
12576:
12191:
5104:
Using the expression from Coulomb's law, we get the total field at
4857:
4853:
4167:
3513:
in vacuum is given by the integral over the distribution of charge
2960:
1359:
1146:
547:
542:
162:
30:
13468:
9147:, in non-relativistic quantum mechanics, the scattering amplitude
4954:
Coulomb's law states that the electric field due to a stationary
1192:
908:
517:
12452:
Cork, C.R. (2015). "Conductive fibres for electronic textiles".
12239:, which uses a similar structure, but for mass instead of charge
4287:
established by other charges that it finds itself in, such that
3939:
is a consequence of historical choices for units. The constant
4481:{\displaystyle |\mathbf {E} |=k_{\text{e}}{\frac {|q|}{r^{2}}}}
3971:
1044:{\displaystyle |F|=k_{\text{e}}{\frac {|q_{1}||q_{2}|}{r^{2}}}}
602:
109:
12757:
The Electrical Researches of the Honourable Henry Cavendish...
13532:
Electric Charges, Polarization, Electric Force, Coulomb's Law
10388:{\displaystyle \mathbf {T} \sin \theta _{1}=\mathbf {F} _{1}}
4845:
1336:
1188:
1169:
1134:
935:
35:
13478:
Sears and Zemansky's University Physics: With Modern Physics
12601:
Acta Helvetica Physico-Mathematico-Anatomico-Botanico-Medica
12388:[Second dissertation on electricity and magnetism].
8477:
can be shown to be a transformation of forces caused by the
4338:
If the field is generated by a positive source point charge
12365:[First dissertation on electricity and magnetism].
9042:
can also be manually verified for the above two equations.
8460:
which is essentially equivalent to Coulomb's law. Thus the
5571:
which is the differential form of Gauss's law, as desired.
4841:
4566:{\textstyle \mathbf {r} _{i}=\mathbf {r} -\mathbf {r} _{i}}
1340:
1233:
1195:
did (i.e., as the inverse square of the distance) included
8156:
4918:
1119:
13507:
10270:. The forces acting on each sphere are three: the weight
9852:
Comparing with the QM scattering, we have to discard the
4861:
2099:
is used for the vector notation. The electrostatic force
13242:
Institute for Theoretical Physics, Heidelberg University
12306:
Halliday, David; Resnick, Robert; Walker, Jearl (2013).
6752:{\displaystyle B_{R}(\mathbf {r} _{0})\subseteq \Omega }
927:. Coulomb's law was essential to the development of the
12896:
The NIST Reference on Constants, Units, and Uncertainty
11413:
11389:
11337:
11318:
2959:
is also used. For a continuous charge distribution, an
13536:
8.02 Electricity and Magnetism, Spring 2002: Lecture 1
13397:"TroisiÚme mémoire sur l'électricité et le magnétisme"
12739:. Vol. 4. London, England: Printed for J. Murray.
11416:
11392:
11340:
11321:
10931:
10904:
10250:, hanging from two ropes of negligible mass of length
9153:
4527:
4364:
4344:
4293:
4271:
4244:
4222:
2886:
2835:
2521:
2457:
2448:
vector at that point, with that point charge removed.
2391:
2288:
2166:
2105:
2069:
2009:
1981:
1869:
1810:
1441:
1331:. In Coulomb's experiment, the torsion balance was an
1203:, both of whom measured the force between plates of a
13453:(6th ed.). New York: W. H. Freeman and Company.
12305:
11946:
11755:
11720:
11691:
11664:
11637:
11308:
11197:
11143:
10980:
10799:
10701:
10667:
10638:
10502:
10422:
10348:
10321:
10299:
10276:
10256:
10236:
10216:
10074:
10048:
9894:
9858:
9593:
9549:
9420:
9217:
9006:
8973:
8951:
8931:
8911:
8889:
8867:
8847:
8649:
8508:
8379:
8277:
8212:
7983:
7961:
7891:
7658:
7559:
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7190:
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6905:
6873:
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6814:
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6765:
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6280:
6185:
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6079:
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6017:
5949:
5914:
5875:
5622:
5589:
5502:
5379:
5281:
5122:
4964:
4926:
Gauss's law § Deriving Gauss's law from Coulomb's law
4579:
4500:
4420:
4007:
3980:
3945:
3813:
3771:
3712:
3660:
3519:
3497:
3477:
3410:
3382:
3355:
3319:
3252:
3224:
3197:
3161:
3142:{\displaystyle dq'=\lambda (\mathbf {r'} )\,d\ell '.}
3090:
3062:
3035:
2999:
2973:
2924:
2864:
2802:
2543:
2499:
2479:
2363:
2326:
2261:
2224:
2134:
2038:
1582:
1553:
1526:
1497:
1470:
958:
13371:"Premier mémoire sur l'électricité et le magnétisme"
12363:"Premier mémoire sur l'électricité et le magnétisme"
12187:
5112:
due to the infinitesimal charge at each other point
1415:
the forces are repulsive (as in the image) and when
1343:
thread. The ball was charged with a known charge of
13528:âa game created by the Molecular Workbench software
13384:"Second mémoire sur l'électricité et le magnétisme"
12698:
Electromagnetics: History, Theory, and Applications
12604:(in Latin). Vol. 4. Basileae. pp. 224â25.
12540:
Electricity and Magnetism: A Historical Perspective
12386:"Second mémoire sur l'électricité et le magnétisme"
11184:{\displaystyle \mathbf {F} _{2}=mg\tan \theta _{2}}
9076:
may be too technical for most readers to understand
7942:{\displaystyle r'_{c}\in \ B_{R}(\mathbf {r} _{0})}
3462:{\displaystyle dq'=\rho ({\boldsymbol {r'}})\,dV'.}
1149:could make a piece of amber attract small objects.
13409:
13028:(3rd ed.). John Wiley & Sons. p. 35.
12928:(Ninth ed.). Cengage Learning. Inside Cover.
12332:
12156:
11905:
11735:
11706:
11677:
11650:
11607:
11277:
11183:
11113:
10961:
10917:
10874:
10764:
10682:
10653:
10608:
10455:
10387:
10329:
10307:
10285:
10262:
10242:
10222:
10117:
10060:
10034:
9880:
9842:
9580:
9531:
9406:
9203:
9018:
8981:
8959:
8937:
8917:
8897:
8875:
8853:
8833:
8636:
8452:
8349:
8261:
8143:
7967:
7941:
7877:
7639:
7543:
7242:
7176:
7034:
6888:
6859:
6820:
6800:
6780:
6751:
6700:
6659:
6594:
6491:
6338:
6264:
6169:
6097:
6065:
6036:
6000:
5935:
5897:
5861:
5608:
5563:
5482:
5354:
5261:
5037:
4819:
4565:
4513:
4480:
4377:
4350:
4319:
4279:
4257:
4230:
4216:. The strength and direction of the Coulomb force
4133:
3993:
3958:
3931:
3800:{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}
3799:
3749:
3697:
3644:
3505:
3483:
3461:
3396:
3368:
3341:
3303:
3238:
3210:
3183:
3141:
3076:
3048:
3021:
2982:
2939:
2910:
2872:
2850:
2815:
2786:
2527:
2507:
2485:
2465:
2418:
2376:
2349:
2312:
2274:
2247:
2199:
2147:
2120:
2091:
2051:
2022:
1994:
1963:
1851:
1794:
1568:
1539:
1512:
1483:
1456:
1435:Coulomb's law states that the electrostatic force
1080:are the quantities of each charge, and the scalar
1043:
12892:"2018 CODATA Value: vacuum electric permittivity"
10962:{\textstyle \mathbf {L} _{2}<\mathbf {L} _{1}}
10690:, assuming Coulomb's law is correct, is equal to
9026:) can be applied to electric currents to get the
2950:
2077:
1339:-coated ball attached to one end, suspended by a
13543:
8905:is the velocity vector of the charged particle,
8026:
3304:{\displaystyle dq'=\sigma (\mathbf {r'} )\,dA'.}
1316:to study the repulsion and attraction forces of
911:that calculates the amount of force between two
10456:{\displaystyle \mathbf {T} \cos \theta _{1}=mg}
2200:{\textstyle \mathbf {F} _{2}=-\mathbf {F} _{1}}
12274:. Hoboken, New Jersey: Wiley. pp. 8, 57.
10969:and the repulsion force between them will be:
8373:points in the radial direction, and so we get
5936:{\displaystyle \mathbf {r} \neq \mathbf {r'} }
3750:{\displaystyle |\mathbf {r} -\mathbf {r'} |=0}
3698:{\displaystyle |\mathbf {r} -\mathbf {r'} |=0}
2357:is negative and the direction of the force on
2255:is positive and the direction of the force on
13475:
13430:
13199:(Third ed.). Cambridge, United Kingdom.
12515:Electrical Stimulation and the Relief of Pain
3349:gives the charge per unit volume at position
3029:gives the charge per unit length at position
2431:
1211:who supposed the inverse-square law in 1758.
877:
13268:
13190:
13188:
12948:: CS1 maint: multiple names: authors list (
12923:
12330:
12310:. John Wiley & Sons. pp. 609, 611.
11929:Using this approximation, the relationship (
10118:{\displaystyle V(r)={\frac {e^{2}}{4\pi r}}}
9633:
9594:
9526:
9487:
9259:
9238:
9195:
9174:
7553:The last equality follows by observing that
5496:" of the Dirac delta function, we arrive at
27:Fundamental physical law of electromagnetism
13480:(13th ed.). Addison-Wesley (Pearson).
13476:Young, Hugh D.; Freedman, Roger A. (2010).
13448:
12966:. CRC Press. 2015-07-28. pp. 465â468.
12807:
12243:Static forces and virtual-particle exchange
10778:
9140:between two charged particles, as follows:
6701:{\displaystyle \mathbf {r} _{0}\in \Omega }
5905:a continuous function (density of charge).
4320:{\textstyle \mathbf {F} =q_{t}\mathbf {E} }
3191:gives the charge per unit area at position
2313:{\textstyle {\widehat {\mathbf {r} }}_{12}}
1133:knew that certain objects, such as rods of
13401:Histoire de l'Académie Royale des Sciences
13388:Histoire de l'Académie Royale des Sciences
13375:Histoire de l'Académie Royale des Sciences
13348:: CS1 maint: location missing publisher (
13301:: CS1 maint: location missing publisher (
13219:: CS1 maint: location missing publisher (
13020:(4th ed.). Prentice Hall. p. 50.
12733:Robison, John (1822). Murray, John (ed.).
12517:. Elsevier Health Sciences. pp. 6â7.
12437:: CS1 maint: location missing publisher (
12390:Histoire de l'Académie Royale des Sciences
12367:Histoire de l'Académie Royale des Sciences
12301:
12299:
6525:
6521:
4871:
1852:{\textstyle \mathbf {r_{12}=r_{1}-r_{2}} }
884:
870:
88:
13437:Fundamentals of the Theory of Electricity
13407:
13194:
13185:
13112:
13084:
13042:Classical Electromagnetism via Relativity
13015:
12803:
12801:
12677:
12652:
12483:
12481:
12408:
12406:
10190:Learn how and when to remove this message
9347:
9104:Learn how and when to remove this message
9088:, without removing the technical details.
6255:
5462:
5439:
5241:
5108:by using an integral to sum the field at
4064:
4060:
4056:
4052:
3444:
3286:
3124:
1191:diminished with distance as the force of
12614:
12562:
12394:History of the Royal Academy of Sciences
12371:History of the Royal Academy of Sciences
12331:Roller, Duane; Roller, D. H. D. (1954).
10201:
9115:
6098:{\displaystyle \Omega \cap V=\emptyset }
4199:
2419:{\textstyle -{\hat {\mathbf {r} }}_{12}}
1358:
1232:
1118:
29:
13394:
13381:
13368:
13145:
13023:
12777:
12749:
12732:
12694:
12537:
12512:
12487:
12412:
12383:
12360:
12356:
12354:
12296:
11935:) becomes the much simpler expression:
9581:{\displaystyle m_{0}\gg |\mathbf {p} |}
9056:
8157:Deriving Coulomb's law from Gauss's law
4919:Deriving Gauss's law from Coulomb's law
3499:
3433:
3404:is an infinitesimal element of volume,
3084:is an infinitesimal element of length,
3022:{\displaystyle \lambda (\mathbf {r} ')}
1304:Finally, in 1785, the French physicist
1141:made the first recorded description of
633:Electromagnetism and special relativity
14:
13544:
13526:A maze game for teaching Coulomb's law
13440:(9th ed.). Moscow: Mir. pp.
13403:. Imprimerie Royale. pp. 612â638.
13390:. Imprimerie Royale. pp. 578â611.
13377:. Imprimerie Royale. pp. 569â577.
13195:Griffiths, David J. (16 August 2018).
13038:
12798:
12781:The Feynman Lectures on Physics Vol II
12478:
12413:Purcell, Edward M. (21 January 2013).
12403:
5609:{\displaystyle \Omega \subseteq R^{3}}
3184:{\displaystyle \sigma (\mathbf {r} ')}
2911:{\textstyle {\hat {\mathbf {r} }}_{i}}
13449:Tipler, Paul A.; Mosca, Gene (2008).
13315:
12773:
12771:
12769:
12767:
12597:
12396:] (in French). pp. 578â611.
12269:
12237:Newton's law of universal gravitation
11685:and the distance between the charges
9086:make it understandable to non-experts
8992:This form of solutions need not obey
4898:Gauss's_law#Relation_to_Coulomb's_law
3246:is an infinitesimal element of area,
2918:is a unit vector in the direction of
2092:{\textstyle \mathbf {\hat {r}} _{12}}
1520:, in the vicinity of another charge,
1228:Newton's law of universal gravitation
1084:is the distance between the charges.
653:Maxwell equations in curved spacetime
13451:Physics for Scientists and Engineers
12926:Physics for Scientists and Engineers
12451:
12377:
12373:] (in French). pp. 569â577.
12351:
12265:
12263:
11937:
11746:
11299:
10971:
10790:
10692:
10493:
10413:
10339:
10142:
9132:> 0), describing electron-proton
9060:
9045:
6860:{\displaystyle \mathbf {E} _{B_{R}}}
6177:and so, for the divergence theorem:
5898:{\displaystyle \rho (\mathbf {r} ')}
4875:
4388:The magnitude of the electric field
3342:{\displaystyle \rho (\mathbf {r} ')}
1354:
1145:around 600 BC, when he noticed that
13395:Coulomb, Charles Augustin (1788) .
13382:Coulomb, Charles Augustin (1788) .
13369:Coulomb, Charles Augustin (1788) .
12808:Fitzpatrick, Richard (2006-02-02).
12750:Maxwell, James Clerk, ed. (1967) .
12490:Intermediate Electromagnetic Theory
10206:Experiment to verify Coulomb's law.
8996:as is the case in the framework of
8861:is the charge of the point source,
4852:and each of the negatively charged
3760:
2967:element of space as a point charge
2858:is the vector from its position to
34:The magnitude of the electrostatic
24:
13522:âa chapter from an online textbook
13362:
12884:
12764:
12462:10.1016/B978-0-08-100201-8.00002-3
9220:
9156:
8088:
7985:
7778:
7659:
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7563:
7478:
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7343:
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7259:
7159:
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7017:
6815:
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6695:
6610:
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6506:
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6419:
6356:
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6236:
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6080:
6057:
5951:
5844:
5810:
5763:
5679:
5590:
5503:
5465:
5380:
5282:
5244:
4186:must be taken into consideration.
4162:The last of these is known as the
4115:
4100:
4091:
1910:
1907:
1904:
1168:("of amber" or "like amber", from
25:
13583:
13495:
13316:Gould, Robert J. (21 July 2020).
13197:Introduction to quantum mechanics
12865:The International System of Units
12736:A System of Mechanical Philosophy
12542:. Greenwood Press. pp. 7â8.
12260:
10061:{\displaystyle \varepsilon \to 0}
7566:
7097:
4189:
3765:The constant of proportionality,
3506:{\displaystyle {\boldsymbol {r}}}
3471:The force on a small test charge
2428:the charges attract each other.
12492:. World Scientific. p. 50.
12190:
11736:{\displaystyle \mathbf {L} _{2}}
11723:
11707:{\displaystyle \mathbf {L} _{1}}
11694:
11146:
10949:
10934:
10683:{\displaystyle \mathbf {F} _{1}}
10670:
10654:{\displaystyle \mathbf {L} _{1}}
10641:
10424:
10375:
10350:
10323:
10301:
10147:
10000:
9991:
9965:
9948:
9936:
9927:
9905:
9829:
9820:
9694:
9685:
9569:
9398:
9386:
9377:
9355:
9343:
9251:
9234:
9187:
9170:
9065:
8975:
8953:
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8806:
8782:
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8651:
8630:
8510:
8468:
8429:
8389:
8381:
8320:
8312:
8298:
8235:
8224:
8128:
8067:
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7994:
7926:
7857:
7821:
7784:
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7742:
7618:
7584:
7537:
7513:
7496:
7466:
7449:
7432:
7402:
7378:
7361:
7331:
7314:
7297:
7230:
7215:
7193:
7165:
7145:
7140:
7137:
7115:
7051:
7023:
7003:
6998:
6995:
6973:
6908:
6889:{\displaystyle \mathbf {E} _{C}}
6876:
6840:
6781:{\displaystyle \mathbf {r} _{0}}
6768:
6730:
6682:
6640:
6636:
6633:
6625:
6615:
6584:
6575:
6551:
6530:
6474:
6454:
6449:
6446:
6430:
6379:
6365:
6296:
6291:
6288:
6245:
6218:
6201:
6127:
6122:
6119:
6037:{\displaystyle V\subseteq R^{3}}
5981:
5977:
5974:
5966:
5956:
5925:
5916:
5884:
5850:
5830:
5825:
5822:
5769:
5741:
5732:
5716:
5707:
5692:
5639:
5625:
5538:
5518:
5510:
5476:
5455:
5447:
5432:
5395:
5387:
5345:
5306:
5295:
5255:
5222:
5214:
5199:
5191:
5180:
5132:
5124:
5013:
4974:
4966:
4924:This section is an excerpt from
4880:
4840:Coulomb's law holds even within
4829:
4793:
4774:
4687:
4662:
4589:
4581:
4553:
4544:
4530:
4427:
4313:
4295:
4273:
4224:
3994:{\displaystyle \varepsilon _{0}}
3959:{\displaystyle \varepsilon _{0}}
3905:
3880:
3816:
3728:
3719:
3676:
3667:
3616:
3607:
3591:
3582:
3529:
3521:
3436:
3358:
3328:
3275:
3200:
3170:
3113:
3038:
3008:
2940:{\displaystyle \mathbf {r} _{i}}
2927:
2892:
2866:
2838:
2757:
2738:
2651:
2626:
2553:
2545:
2501:
2459:
2400:
2294:
2187:
2169:
2108:
2074:
2052:{\displaystyle \varepsilon _{0}}
1943:
1925:
1875:
1843:
1839:
1835:
1830:
1826:
1822:
1817:
1813:
1768:
1749:
1674:
1649:
1585:
1569:{\displaystyle \mathbf {r} _{2}}
1556:
1513:{\displaystyle \mathbf {r} _{1}}
1500:
1444:
13416:(3rd ed.). Prentice Hall.
13412:Introduction to Electrodynamics
13309:
13262:
13251:from the original on 2021-05-04
13227:
13174:from the original on 2023-12-30
13139:
13115:Introduction to electrodynamics
13106:
13095:from the original on 2022-10-09
13078:
13067:from the original on 2022-10-09
13032:
13018:Introduction to Electrodynamics
13008:
12956:
12917:
12906:from the original on 2016-06-03
12856:
12845:from the original on 2014-10-10
12827:
12816:from the original on 2015-07-09
12743:
12726:
12715:from the original on 2014-03-10
12688:
12684:. London, England. p. 732.
12671:
12646:
12608:
12591:
12556:
12531:
10138:
9414:This is to be compared to the:
8202:, centered at the point charge
4848:between the positively charged
2955:In this case, the principle of
13152:. Cambridge University Press.
13146:Purcell, Edward (2011-09-22).
12506:
12445:
12324:
12112:
12038:
11813:
11515:
11011:
10998:
10568:
10084:
10078:
10052:
10010:
9986:
9944:
9923:
9909:
9901:
9869:
9859:
9837:
9816:
9804:
9794:
9791:
9748:
9736:
9726:
9704:
9680:
9638:
9620:
9612:
9574:
9564:
9513:
9505:
9452:
9442:
9394:
9373:
9359:
9351:
9326:
9317:
9314:
9283:
9262:
9245:
9241:
9229:
9225:
9198:
9181:
9177:
9165:
9161:
9128:admits continuum states (with
8750:
8714:
8609:
8573:
8433:
8393:
8385:
8324:
8316:
8302:
8138:
8123:
8097:
8091:
8081:
8077:
8062:
8048:
8033:
8019:
8004:
7936:
7921:
7871:
7867:
7852:
7838:
7834:
7816:
7773:
7760:
7752:
7737:
7686:
7680:
7668:
7662:
7628:
7613:
7597:
7594:
7579:
7560:
7506:
7491:
7442:
7427:
7371:
7356:
7307:
7292:
7268:
7262:
7154:
7133:
7125:
7110:
7012:
6991:
6983:
6968:
6740:
6725:
6648:
6629:
6522:
6463:
6442:
6383:
6375:
6333:
6321:
6305:
6284:
6164:
6152:
6136:
6115:
5989:
5970:
5892:
5879:
5839:
5818:
5750:
5727:
5700:
5687:
5643:
5635:
5542:
5534:
5522:
5514:
5459:
5443:
5436:
5428:
5399:
5391:
5349:
5341:
5312:
5301:
5228:
5209:
5203:
5187:
5184:
5176:
5136:
5128:
4978:
4970:
4804:
4787:
4698:
4681:
4666:
4593:
4585:
4461:
4453:
4432:
4422:
4410:at a certain distance from it
4265:depends on the electric field
4143:
4071:
4065:
3916:
3899:
3884:
3737:
3714:
3706:that is not supposed to allow
3685:
3662:
3626:
3602:
3533:
3525:
3441:
3428:
3336:
3323:
3283:
3270:
3178:
3165:
3121:
3108:
3016:
3003:
2951:Continuous charge distribution
2896:
2768:
2751:
2662:
2645:
2630:
2557:
2549:
2535:discrete charges in vacuum is
2404:
2211:If both charges have the same
1954:
1937:
1879:
1779:
1762:
1685:
1668:
1653:
1024:
1009:
1004:
989:
968:
960:
13:
1:
13271:Lectures on quantum mechanics
13235:"Quantum Field Theory I + II"
12653:Schofield, Robert E. (1997).
12254:
10337:. In the equilibrium state:
6808:as radius (it exists because
3369:{\displaystyle \mathbf {r} '}
3211:{\displaystyle \mathbf {r} '}
3049:{\displaystyle \mathbf {r} '}
2851:{\textstyle \mathbf {r} _{i}}
2121:{\textstyle \mathbf {F} _{2}}
1457:{\textstyle \mathbf {F} _{1}}
658:Relativistic electromagnetism
13408:Griffiths, David J. (1999).
13113:Griffiths, David J. (1999).
13024:Jackson, John David (1999).
13016:Griffiths, David J. (2013).
12778:Feynman, Richard P. (1970).
12339:. Cambridge, Massachusetts:
10330:{\displaystyle \mathbf {F} }
10308:{\displaystyle \mathbf {T} }
10172:Knowledge's inclusion policy
8982:{\displaystyle \mathbf {v} }
8960:{\displaystyle \mathbf {r} }
8898:{\displaystyle \mathbf {v} }
8876:{\displaystyle \mathbf {r} }
5869:be the electric field, with
5616:be a bounded open set, and
3968:vacuum electric permittivity
2873:{\displaystyle \mathbf {r} }
2508:{\displaystyle \mathbf {r} }
1390:is the force experienced by
1372:is the force experienced by
1244:In 1769, Scottish physicist
1129:Ancient cultures around the
1101:law of universal gravitation
897:Coulomb's inverse-square law
7:
12695:Elliott, Robert S. (1999).
12417:(3rd ed.). Cambridge.
12183:
12170:
11931:
11919:
11678:{\displaystyle \theta _{2}}
11651:{\displaystyle \theta _{1}}
11621:
11293:
11287:
11127:
10918:{\textstyle {\frac {q}{2}}}
10888:
10622:
10487:
10481:
10469:
10401:
9019:{\displaystyle \beta \ll 1}
6011:Consider now a compact set
5580:Proof (without Dirac Delta)
4939:only. However, Gauss's law
4894:the scope of other articles
4844:, correctly describing the
4403:created by a single source
4164:electrostatic approximation
4001:, the Coulomb constant is
3974:2018 recommended value for
1306:Charles-Augustin de Coulomb
1152:In 1600, English scientist
1124:Charles-Augustin de Coulomb
925:Charles-Augustin de Coulomb
10:
13588:
13085:Heaviside, Oliver (1894).
12678:Priestley, Joseph (1767).
10125:as the Coulomb potential.
9049:
9036:LiĂ©nardâWiechert potential
8179:). However, Coulomb's law
7647:, and the argument above.
6759:as the sphere centered in
6066:{\displaystyle \partial V}
4923:
4833:
4333:principle of superposition
4193:
2432:System of discrete charges
2350:{\displaystyle q_{1}q_{2}}
2248:{\displaystyle q_{1}q_{2}}
1576:, in a vacuum is equal to
1310:theory of electromagnetism
1214:Based on experiments with
1170:
1114:
929:theory of electromagnetism
383:LiĂ©nardâWiechert potential
13562:Eponymous laws of physics
13318:Electromagnetic processes
13149:Electricity and Magnetism
13051:10.1007/978-1-4899-6559-2
13039:Rosser, W. G. V. (1968).
13026:Classical Electrodynamics
12632:(including footnote 44).
12415:Electricity and magnetism
4280:{\textstyle \mathbf {E} }
4231:{\textstyle \mathbf {F} }
2466:{\textstyle \mathbf {F} }
1464:experienced by a charge,
1363:In the image, the vector
648:Mathematical descriptions
358:Electromagnetic radiation
348:Electromagnetic induction
288:Magnetic vector potential
283:Magnetic scalar potential
13513:Electricity and the Atom
13158:10.1017/cbo9781139005043
12488:Stewart, Joseph (2001).
12341:Harvard University Press
9881:{\displaystyle (2m)^{2}}
6499:for the argument above (
2823:is the magnitude of the
2215:(like charges) then the
1095:, the law is similar to
12812:. University of Texas.
12538:Baigrie, Brian (2007).
12513:Simpson, Brian (2003).
12308:Fundamentals of Physics
12270:Huray, Paul G. (2010).
12227:Method of image charges
10315:and the electric force
8938:{\displaystyle \theta }
8173:Helmholtz decomposition
6821:{\displaystyle \Omega }
4945:superposition principle
4872:Relation to Gauss's law
4208:An electric field is a
3077:{\displaystyle d\ell '}
198:Electrostatic induction
193:Electrostatic discharge
12158:
11907:
11737:
11708:
11679:
11652:
11609:
11279:
11185:
11115:
10963:
10919:
10876:
10766:
10684:
10655:
10610:
10457:
10389:
10331:
10309:
10287:
10264:
10244:
10224:
10207:
10119:
10068:at the end will yield
10062:
10036:
9882:
9844:
9582:
9533:
9408:
9205:
9138:non-relativistic limit
9121:
9034:given by solutions of
9020:
8983:
8961:
8939:
8919:
8918:{\displaystyle \beta }
8899:
8877:
8855:
8835:
8638:
8492:Lorentz transformation
8454:
8351:
8263:
8145:
7969:
7955:and the continuity of
7943:
7879:
7641:
7545:
7244:
7178:
7036:
6890:
6861:
6822:
6802:
6782:
6753:
6702:
6661:
6596:
6493:
6340:
6266:
6171:
6099:
6067:
6038:
6002:
5937:
5899:
5863:
5610:
5565:
5484:
5356:
5263:
5039:
4821:
4759:
4644:
4567:
4515:
4482:
4414:in vacuum is given by
4379:
4352:
4321:
4281:
4259:
4232:
4205:
4135:
3995:
3960:
3933:
3801:
3751:
3699:
3646:
3507:
3485:
3463:
3398:
3370:
3343:
3305:
3240:
3212:
3185:
3143:
3078:
3050:
3023:
2984:
2941:
2912:
2874:
2852:
2817:
2788:
2723:
2608:
2529:
2509:
2487:
2467:
2420:
2378:
2351:
2314:
2276:
2249:
2201:
2149:
2122:
2093:
2053:
2024:
1996:
1965:
1853:
1796:
1570:
1541:
1514:
1485:
1458:
1432:
1241:
1126:
1045:
950:
628:Electromagnetic tensor
68:
13269:Baym, Gordon (2018).
12565:Philosophy of Science
12217:Electromagnetic force
12159:
11908:
11738:
11709:
11680:
11653:
11631:Measuring the angles
11610:
11280:
11186:
11116:
10964:
10920:
10877:
10767:
10685:
10656:
10611:
10458:
10390:
10332:
10310:
10288:
10265:
10245:
10230:and same-sign charge
10225:
10205:
10120:
10063:
10037:
9883:
9845:
9583:
9534:
9409:
9206:
9119:
9021:
8984:
8962:
8945:is the angle between
8940:
8920:
8900:
8878:
8856:
8836:
8639:
8455:
8352:
8264:
8146:
7970:
7968:{\displaystyle \rho }
7944:
7880:
7642:
7546:
7245:
7179:
7037:
6891:
6862:
6823:
6803:
6783:
6754:
6703:
6662:
6597:
6494:
6341:
6267:
6172:
6100:
6068:
6039:
6003:
5938:
5900:
5864:
5611:
5566:
5485:
5357:
5264:
5040:
4906:and help introduce a
4822:
4739:
4624:
4568:
4516:
4514:{\displaystyle q_{i}}
4483:
4380:
4353:
4322:
4282:
4260:
4233:
4203:
4136:
3996:
3961:
3934:
3802:
3752:
3700:
3647:
3508:
3486:
3464:
3399:
3371:
3344:
3306:
3241:
3213:
3186:
3144:
3079:
3051:
3024:
2985:
2942:
2913:
2875:
2853:
2818:
2816:{\displaystyle q_{i}}
2789:
2703:
2588:
2530:
2515:, due to a system of
2510:
2488:
2468:
2421:
2379:
2377:{\displaystyle q_{1}}
2352:
2315:
2277:
2275:{\displaystyle q_{1}}
2250:
2202:
2150:
2148:{\displaystyle q_{2}}
2123:
2094:
2054:
2025:
1997:
1966:
1863:between the charges,
1854:
1797:
1571:
1542:
1540:{\displaystyle q_{2}}
1515:
1486:
1484:{\displaystyle q_{1}}
1459:
1362:
1236:
1181:Pseudodoxia Epidemica
1122:
1046:
945:
903:, is an experimental
621:Covariant formulation
413:Synchrotron radiation
353:Electromagnetic pulse
343:Electromagnetic field
33:
12598:Socin, Abel (1760).
11944:
11753:
11718:
11689:
11662:
11635:
11306:
11195:
11141:
10978:
10929:
10902:
10797:
10699:
10665:
10636:
10500:
10420:
10346:
10319:
10297:
10274:
10254:
10234:
10214:
10072:
10046:
9892:
9856:
9591:
9547:
9418:
9215:
9151:
9057:Quantum field theory
9004:
8971:
8949:
8929:
8909:
8887:
8865:
8845:
8647:
8506:
8377:
8275:
8210:
7981:
7959:
7889:
7656:
7557:
7256:
7188:
7046:
6903:
6871:
6835:
6812:
6792:
6763:
6712:
6677:
6606:
6503:
6352:
6278:
6183:
6109:
6077:
6054:
6015:
5947:
5912:
5873:
5620:
5587:
5500:
5377:
5371:Dirac delta function
5279:
5120:
4962:
4577:
4525:
4498:
4418:
4362:
4342:
4291:
4269:
4242:
4220:
4184:theory of relativity
4005:
3978:
3943:
3811:
3807:, in Coulomb's law:
3769:
3710:
3658:
3517:
3495:
3475:
3408:
3380:
3353:
3317:
3250:
3222:
3195:
3159:
3088:
3060:
3033:
2997:
2971:
2957:linear superposition
2922:
2884:
2862:
2833:
2800:
2541:
2519:
2497:
2477:
2455:
2438:law of superposition
2389:
2361:
2324:
2286:
2259:
2222:
2164:
2132:
2103:
2067:
2036:
2007:
1979:
1867:
1808:
1580:
1551:
1524:
1495:
1468:
1439:
1216:electrically charged
956:
913:electrically charged
663:Stressâenergy tensor
588:Reluctance (complex)
333:Displacement current
12454:Electronic Textiles
12272:Maxwell's equations
12232:Molecular modelling
11448:
11415:
11391:
11372:
11339:
11320:
11246:
11107:
11052:
10843:
10758:
10293:, the rope tension
9040:Maxwell's equations
9032:Maxwell's equations
7904:
7833:
5908:It is true for all
4176:Maxwell's equations
3397:{\displaystyle dV'}
3239:{\displaystyle dA'}
1861:displacement vector
917:electrostatic force
578:Magnetomotive force
463:Electromotive force
433:Alternating current
368:Jefimenko equations
328:Cyclotron radiation
13518:2009-02-21 at the
13320:. Princeton, N.J.
13045:. pp. 29â42.
13014:See, for example,
12972:10.1201/b18636-105
12784:. Addison-Wesley.
12198:Electronics portal
12154:
11903:
11733:
11704:
11675:
11648:
11605:
11450:
11434:
11410:
11374:
11358:
11334:
11275:
11232:
11181:
11111:
11093:
11038:
10959:
10915:
10872:
10829:
10762:
10744:
10680:
10651:
10606:
10453:
10385:
10327:
10305:
10286:{\displaystyle mg}
10283:
10260:
10240:
10220:
10208:
10115:
10058:
10032:
9878:
9840:
9578:
9529:
9404:
9201:
9145:Born approximation
9122:
9052:Electric potential
9016:
8998:special relativity
8994:Newton's third law
8979:
8957:
8935:
8915:
8895:
8873:
8851:
8831:
8634:
8483:Maxwell's equation
8462:inverse-square law
8450:
8347:
8259:
8192:
8141:
8040:
7975:, one arrives at:
7965:
7939:
7892:
7875:
7819:
7637:
7541:
7240:
7174:
7032:
6886:
6857:
6818:
6798:
6778:
6749:
6698:
6657:
6592:
6489:
6336:
6262:
6167:
6105:. It follows that
6095:
6063:
6034:
5998:
5933:
5895:
5859:
5606:
5581:
5561:
5480:
5352:
5259:
5116:in space, to give
5035:
4952:
4904:discuss this issue
4817:
4563:
4511:
4478:
4378:{\textstyle q_{t}}
4375:
4348:
4317:
4277:
4258:{\textstyle q_{t}}
4255:
4228:
4206:
4131:
3991:
3956:
3929:
3797:
3747:
3695:
3642:
3503:
3481:
3459:
3394:
3366:
3339:
3301:
3236:
3208:
3181:
3139:
3074:
3046:
3019:
2983:{\displaystyle dq}
2980:
2937:
2908:
2870:
2848:
2813:
2784:
2525:
2505:
2483:
2473:on a small charge
2463:
2416:
2374:
2347:
2310:
2272:
2245:
2197:
2157:Newton's third law
2145:
2118:
2089:
2049:
2023:{\textstyle q_{1}}
2020:
1995:{\textstyle q_{2}}
1992:
1961:
1849:
1792:
1566:
1537:
1510:
1481:
1454:
1433:
1345:static electricity
1242:
1224:inverse-square law
1143:static electricity
1127:
1109:tested extensively
1099:'s inverse-square
1093:inverse-square law
1041:
938:between two point
426:Electrical network
263:Gauss magnetic law
228:Static electricity
188:Electric potential
69:
13487:978-0-321-69686-1
13460:978-0-7167-8964-2
13423:978-0-13-805326-0
13327:978-0-691-21584-6
13280:978-0-429-49926-5
13206:978-1-107-18963-8
13167:978-1-107-01360-5
13060:978-1-4899-6258-4
12981:978-0-429-22704-2
12935:978-1-133-95405-7
12878:978-92-822-2272-0
12708:978-0-7803-5384-8
12664:978-0-271-01662-7
12549:978-0-313-33358-3
12524:978-0-444-51258-1
12499:978-981-02-4471-2
12281:978-0-470-54991-9
12212:Darwin Lagrangian
12178:
12177:
12152:
12137:
12099:
12063:
12025:
11989:
11988:
11968:
11927:
11926:
11901:
11900:
11880:
11854:
11811:
11793:
11789:
11629:
11628:
11603:
11549:
11513:
11458:
11452:
11414:
11408:
11390:
11376:
11338:
11319:
11248:
11214:
11135:
11134:
11109:
11075:
11054:
11009:
10913:
10896:
10895:
10845:
10786:
10785:
10760:
10630:
10629:
10566:
10541:
10477:
10476:
10409:
10408:
10263:{\displaystyle l}
10243:{\displaystyle q}
10223:{\displaystyle m}
10200:
10199:
10192:
10113:
10030:
9724:
9462:
9126:Coulomb potential
9114:
9113:
9106:
9046:Coulomb potential
8854:{\displaystyle q}
8829:
8797:
8768:
8691:
8627:
8550:
8487:Lorentz force law
8448:
8436:
8422:
8345:
8305:
8257:
8190:
8118:
8086:
8025:
7910:
7811:
7720:
7700:
7087:
6951:
6828:is an open set).
6801:{\displaystyle R}
6573:
6570:
6567:
6545:
6542:
6412:
5803:
5760:
5672:
5579:
5556:
5420:
5323:
5239:
5165:
5088:electric constant
5033:
5007:
4950:
4916:
4915:
4836:Coulomb explosion
4815:
4737:
4709:
4669:
4622:
4494:discrete charges
4476:
4446:
4089:
4044:
4015:
3927:
3887:
3871:
3795:
3637:
3562:
3484:{\displaystyle q}
2899:
2779:
2701:
2673:
2633:
2586:
2486:{\displaystyle q}
2407:
2301:
2080:
2061:electric constant
1959:
1920:
1915:
1893:
1882:
1790:
1743:
1696:
1656:
1640:
1381:, and the vector
1355:Mathematical form
1318:charged particles
1139:Thales of Miletus
1039:
982:
894:
893:
593:Reluctance (real)
563:Gyratorâcapacitor
508:Resonant cavities
398:Maxwell equations
16:(Redirected from
13579:
13552:Electromagnetism
13491:
13472:
13445:
13427:
13415:
13404:
13391:
13378:
13354:
13353:
13347:
13339:
13313:
13307:
13306:
13300:
13292:
13266:
13260:
13259:
13257:
13256:
13250:
13239:
13231:
13225:
13224:
13218:
13210:
13192:
13183:
13182:
13180:
13179:
13143:
13137:
13136:
13110:
13104:
13103:
13101:
13100:
13082:
13076:
13075:
13073:
13072:
13036:
13030:
13029:
13021:
13012:
13006:
13005:
12999:
12995:
12993:
12985:
12960:
12954:
12953:
12947:
12939:
12921:
12915:
12914:
12912:
12911:
12888:
12882:
12881:
12870:
12860:
12854:
12853:
12851:
12850:
12831:
12825:
12824:
12822:
12821:
12805:
12796:
12795:
12775:
12762:
12761:
12747:
12741:
12740:
12730:
12724:
12723:
12721:
12720:
12692:
12686:
12685:
12675:
12669:
12668:
12650:
12644:
12643:
12612:
12606:
12605:
12595:
12589:
12588:
12560:
12554:
12553:
12535:
12529:
12528:
12510:
12504:
12503:
12485:
12476:
12475:
12449:
12443:
12442:
12436:
12428:
12410:
12401:
12400:
12384:Coulomb (1785).
12381:
12375:
12374:
12361:Coulomb (1785).
12358:
12349:
12348:
12338:
12328:
12322:
12321:
12303:
12294:
12293:
12267:
12200:
12195:
12194:
12172:
12163:
12161:
12160:
12155:
12153:
12151:
12143:
12138:
12136:
12135:
12126:
12125:
12116:
12111:
12110:
12105:
12104:
12100:
12098:
12097:
12088:
12087:
12078:
12064:
12062:
12061:
12052:
12051:
12042:
12037:
12036:
12031:
12030:
12026:
12024:
12023:
12014:
12013:
12004:
11990:
11987:
11979:
11978:
11969:
11967:
11959:
11958:
11949:
11948:
11938:
11921:
11912:
11910:
11909:
11904:
11902:
11899:
11891:
11890:
11881:
11879:
11871:
11870:
11861:
11860:
11855:
11853:
11852:
11851:
11835:
11834:
11833:
11817:
11812:
11810:
11799:
11794:
11782:
11781:
11747:
11742:
11740:
11739:
11734:
11732:
11731:
11726:
11713:
11711:
11710:
11705:
11703:
11702:
11697:
11684:
11682:
11681:
11676:
11674:
11673:
11657:
11655:
11654:
11649:
11647:
11646:
11623:
11614:
11612:
11611:
11606:
11604:
11602:
11601:
11600:
11584:
11583:
11582:
11566:
11561:
11560:
11555:
11554:
11550:
11548:
11547:
11538:
11537:
11528:
11514:
11512:
11511:
11510:
11488:
11487:
11486:
11464:
11459:
11457:
11453:
11451:
11449:
11447:
11442:
11433:
11432:
11411:
11409:
11404:
11403:
11394:
11387:
11381:
11377:
11375:
11373:
11371:
11366:
11357:
11356:
11335:
11333:
11332:
11331:
11316:
11310:
11300:
11284:
11282:
11281:
11276:
11274:
11273:
11249:
11247:
11245:
11240:
11231:
11230:
11210:
11209:
11200:
11199:
11190:
11188:
11187:
11182:
11180:
11179:
11155:
11154:
11149:
11129:
11120:
11118:
11117:
11112:
11110:
11108:
11106:
11101:
11092:
11091:
11071:
11070:
11061:
11060:
11055:
11053:
11051:
11046:
11037:
11036:
11020:
11019:
11014:
11010:
11002:
10995:
10990:
10989:
10972:
10968:
10966:
10965:
10960:
10958:
10957:
10952:
10943:
10942:
10937:
10924:
10922:
10921:
10916:
10914:
10906:
10890:
10881:
10879:
10878:
10873:
10871:
10870:
10846:
10844:
10842:
10837:
10828:
10827:
10811:
10810:
10801:
10791:
10780:
10771:
10769:
10768:
10763:
10761:
10759:
10757:
10752:
10743:
10742:
10726:
10725:
10716:
10711:
10710:
10693:
10689:
10687:
10686:
10681:
10679:
10678:
10673:
10660:
10658:
10657:
10652:
10650:
10649:
10644:
10624:
10615:
10613:
10612:
10607:
10605:
10604:
10580:
10579:
10567:
10565:
10557:
10556:
10547:
10542:
10540:
10539:
10538:
10522:
10521:
10520:
10504:
10494:
10471:
10462:
10460:
10459:
10454:
10443:
10442:
10427:
10414:
10403:
10394:
10392:
10391:
10386:
10384:
10383:
10378:
10369:
10368:
10353:
10340:
10336:
10334:
10333:
10328:
10326:
10314:
10312:
10311:
10306:
10304:
10292:
10290:
10289:
10284:
10269:
10267:
10266:
10261:
10249:
10247:
10246:
10241:
10229:
10227:
10226:
10221:
10195:
10188:
10184:
10181:
10175:
10151:
10150:
10143:
10133:Yukawa potential
10124:
10122:
10121:
10116:
10114:
10112:
10101:
10100:
10091:
10067:
10065:
10064:
10059:
10041:
10039:
10038:
10033:
10031:
10029:
10019:
10018:
10013:
10007:
10003:
9994:
9989:
9983:
9982:
9973:
9968:
9963:
9962:
9953:
9952:
9951:
9943:
9939:
9930:
9908:
9887:
9885:
9884:
9879:
9877:
9876:
9849:
9847:
9846:
9841:
9836:
9832:
9823:
9812:
9811:
9790:
9789:
9782:
9766:
9765:
9744:
9743:
9725:
9723:
9713:
9712:
9707:
9701:
9697:
9688:
9683:
9677:
9676:
9667:
9656:
9655:
9641:
9623:
9615:
9604:
9587:
9585:
9584:
9579:
9577:
9572:
9567:
9559:
9558:
9538:
9536:
9535:
9530:
9516:
9508:
9497:
9486:
9485:
9484:
9483:
9463:
9461:
9460:
9459:
9440:
9436:
9435:
9425:
9413:
9411:
9410:
9405:
9403:
9402:
9401:
9393:
9389:
9380:
9358:
9346:
9341:
9340:
9313:
9312:
9311:
9295:
9294:
9258:
9254:
9248:
9237:
9232:
9224:
9223:
9210:
9208:
9207:
9202:
9194:
9190:
9184:
9173:
9168:
9160:
9159:
9109:
9102:
9098:
9095:
9089:
9069:
9068:
9061:
9025:
9023:
9022:
9017:
8988:
8986:
8985:
8980:
8978:
8966:
8964:
8963:
8958:
8956:
8944:
8942:
8941:
8936:
8924:
8922:
8921:
8916:
8904:
8902:
8901:
8896:
8894:
8882:
8880:
8879:
8874:
8872:
8860:
8858:
8857:
8852:
8840:
8838:
8837:
8832:
8830:
8828:
8827:
8818:
8817:
8809:
8803:
8798:
8796:
8795:
8786:
8785:
8777:
8771:
8769:
8767:
8766:
8765:
8761:
8742:
8741:
8732:
8731:
8712:
8711:
8710:
8694:
8692:
8690:
8689:
8688:
8679:
8678:
8659:
8654:
8643:
8641:
8640:
8635:
8633:
8628:
8626:
8625:
8624:
8620:
8601:
8600:
8591:
8590:
8571:
8570:
8569:
8553:
8551:
8549:
8548:
8547:
8538:
8537:
8518:
8513:
8459:
8457:
8456:
8451:
8449:
8447:
8446:
8437:
8432:
8427:
8425:
8423:
8421:
8420:
8419:
8400:
8392:
8384:
8372:
8362:
8356:
8354:
8353:
8348:
8346:
8344:
8343:
8331:
8323:
8315:
8307:
8306:
8301:
8296:
8293:
8292:
8268:
8266:
8265:
8260:
8258:
8256:
8255:
8243:
8238:
8227:
8222:
8221:
8205:
8201:
8197:
8191:Outline of proof
8170:
8150:
8148:
8147:
8142:
8137:
8136:
8131:
8119:
8117:
8116:
8104:
8087:
8085:
8084:
8076:
8075:
8070:
8061:
8060:
8051:
8042:
8039:
8018:
8017:
8012:
8003:
8002:
7997:
7988:
7974:
7972:
7971:
7966:
7948:
7946:
7945:
7940:
7935:
7934:
7929:
7920:
7919:
7908:
7900:
7884:
7882:
7881:
7876:
7874:
7866:
7865:
7860:
7851:
7850:
7841:
7829:
7824:
7812:
7810:
7809:
7797:
7792:
7791:
7787:
7781:
7772:
7768:
7756:
7755:
7751:
7750:
7745:
7736:
7735:
7721:
7719:
7718:
7706:
7701:
7699:
7698:
7689:
7675:
7646:
7644:
7643:
7638:
7627:
7626:
7621:
7612:
7611:
7593:
7592:
7587:
7578:
7577:
7550:
7548:
7547:
7542:
7540:
7529:
7528:
7527:
7526:
7516:
7510:
7509:
7505:
7504:
7499:
7490:
7489:
7469:
7458:
7457:
7452:
7446:
7445:
7441:
7440:
7435:
7426:
7425:
7405:
7394:
7393:
7392:
7391:
7381:
7375:
7374:
7370:
7369:
7364:
7355:
7354:
7334:
7323:
7322:
7317:
7311:
7310:
7306:
7305:
7300:
7291:
7290:
7249:
7247:
7246:
7241:
7239:
7238:
7233:
7224:
7223:
7218:
7209:
7208:
7207:
7206:
7196:
7183:
7181:
7180:
7175:
7173:
7172:
7168:
7162:
7153:
7152:
7148:
7129:
7128:
7124:
7123:
7118:
7109:
7108:
7088:
7086:
7085:
7084:
7065:
7060:
7059:
7054:
7041:
7039:
7038:
7033:
7031:
7030:
7026:
7020:
7011:
7010:
7006:
6987:
6986:
6982:
6981:
6976:
6967:
6966:
6952:
6950:
6949:
6948:
6929:
6924:
6923:
6922:
6921:
6911:
6895:
6893:
6892:
6887:
6885:
6884:
6879:
6866:
6864:
6863:
6858:
6856:
6855:
6854:
6853:
6843:
6827:
6825:
6824:
6819:
6807:
6805:
6804:
6799:
6787:
6785:
6784:
6779:
6777:
6776:
6771:
6758:
6756:
6755:
6750:
6739:
6738:
6733:
6724:
6723:
6707:
6705:
6704:
6699:
6691:
6690:
6685:
6666:
6664:
6663:
6658:
6647:
6646:
6628:
6620:
6619:
6618:
6601:
6599:
6598:
6593:
6591:
6590:
6578:
6571:
6568:
6565:
6558:
6557:
6543:
6540:
6533:
6498:
6496:
6495:
6490:
6482:
6481:
6477:
6471:
6462:
6461:
6457:
6435:
6434:
6433:
6423:
6422:
6413:
6411:
6410:
6409:
6390:
6382:
6374:
6373:
6368:
6359:
6345:
6343:
6342:
6337:
6320:
6319:
6304:
6303:
6299:
6271:
6269:
6268:
6263:
6254:
6253:
6248:
6239:
6234:
6233:
6221:
6210:
6209:
6204:
6198:
6197:
6176:
6174:
6173:
6168:
6151:
6150:
6135:
6134:
6130:
6104:
6102:
6101:
6096:
6072:
6070:
6069:
6064:
6043:
6041:
6040:
6035:
6033:
6032:
6007:
6005:
6004:
5999:
5988:
5987:
5969:
5961:
5960:
5959:
5942:
5940:
5939:
5934:
5932:
5931:
5919:
5904:
5902:
5901:
5896:
5891:
5887:
5868:
5866:
5865:
5860:
5858:
5857:
5853:
5847:
5838:
5837:
5833:
5814:
5813:
5804:
5802:
5801:
5800:
5781:
5776:
5772:
5766:
5761:
5759:
5758:
5753:
5749:
5748:
5744:
5735:
5724:
5723:
5719:
5710:
5704:
5699:
5695:
5683:
5682:
5673:
5671:
5670:
5669:
5650:
5642:
5634:
5633:
5628:
5615:
5613:
5612:
5607:
5605:
5604:
5570:
5568:
5567:
5562:
5557:
5555:
5554:
5545:
5541:
5529:
5521:
5513:
5494:sifting property
5489:
5487:
5486:
5481:
5479:
5474:
5473:
5468:
5458:
5450:
5435:
5421:
5419:
5418:
5406:
5398:
5390:
5373:, the result is
5368:
5361:
5359:
5358:
5353:
5348:
5328:
5324:
5322:
5321:
5320:
5315:
5309:
5304:
5298:
5293:
5272:
5268:
5266:
5265:
5260:
5258:
5253:
5252:
5247:
5240:
5238:
5237:
5236:
5231:
5225:
5217:
5212:
5206:
5202:
5194:
5183:
5171:
5166:
5164:
5163:
5162:
5143:
5135:
5127:
5115:
5111:
5107:
5095:
5085:
5074:
5072:
5066:
5056:
5044:
5042:
5041:
5036:
5034:
5032:
5031:
5022:
5021:
5016:
5010:
5008:
5006:
5005:
5004:
4985:
4977:
4969:
4951:Outline of proof
4911:
4884:
4883:
4876:
4826:
4824:
4823:
4818:
4816:
4814:
4813:
4808:
4807:
4802:
4801:
4796:
4790:
4783:
4782:
4777:
4771:
4769:
4768:
4758:
4753:
4738:
4736:
4735:
4734:
4715:
4710:
4708:
4707:
4702:
4701:
4696:
4695:
4690:
4684:
4677:
4676:
4671:
4670:
4665:
4660:
4656:
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4638:
4623:
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4592:
4584:
4572:
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4547:
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4520:
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4512:
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4509:
4487:
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4479:
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4465:
4464:
4456:
4450:
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4435:
4430:
4425:
4402:
4393:
4384:
4382:
4381:
4376:
4374:
4373:
4357:
4355:
4354:
4349:
4326:
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4318:
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4311:
4310:
4298:
4286:
4284:
4283:
4278:
4276:
4264:
4262:
4261:
4256:
4254:
4253:
4237:
4235:
4234:
4229:
4227:
4214:unit test charge
4140:
4138:
4137:
4132:
4127:
4126:
4125:
4113:
4108:
4107:
4098:
4087:
4086:
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4022:
4017:
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4000:
3998:
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3990:
3989:
3965:
3963:
3962:
3957:
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3938:
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3824:
3819:
3806:
3804:
3803:
3798:
3796:
3794:
3793:
3792:
3773:
3761:Coulomb constant
3757:to be analyzed.
3756:
3754:
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3748:
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3722:
3717:
3704:
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3015:
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2279:
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2254:
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2251:
2246:
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2234:
2233:
2208:
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2203:
2198:
2196:
2195:
2190:
2178:
2177:
2172:
2154:
2152:
2151:
2146:
2144:
2143:
2127:
2125:
2124:
2119:
2117:
2116:
2111:
2098:
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2073:
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2019:
2018:
2001:
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1960:
1958:
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1952:
1951:
1946:
1940:
1934:
1933:
1928:
1922:
1918:
1917:
1916:
1914:
1913:
1901:
1896:
1891:
1890:
1889:
1884:
1883:
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1873:
1858:
1856:
1855:
1850:
1848:
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1799:
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1789:
1788:
1783:
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1777:
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1752:
1746:
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1652:
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1565:
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1559:
1546:
1544:
1543:
1538:
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1519:
1517:
1516:
1511:
1509:
1508:
1503:
1490:
1488:
1487:
1482:
1480:
1479:
1463:
1461:
1460:
1455:
1453:
1452:
1447:
1430:
1414:
1398:
1389:
1380:
1371:
1300:
1298:
1296:
1295:
1292:
1289:
1281:and that of the
1280:
1278:
1276:
1275:
1272:
1269:
1253:
1220:Joseph Priestley
1201:Alessandro Volta
1197:Daniel Bernoulli
1173:
1172:
1079:
1070:
1061:
1050:
1048:
1047:
1042:
1040:
1038:
1037:
1028:
1027:
1022:
1021:
1012:
1007:
1002:
1001:
992:
986:
984:
983:
980:
971:
963:
886:
879:
872:
553:Electric machine
536:Magnetic circuit
498:Parallel circuit
488:Network analysis
453:Electric current
388:London equations
233:Triboelectricity
223:Potential energy
92:
82:Electromagnetism
73:
72:
65:
56:
44:
21:
13587:
13586:
13582:
13581:
13580:
13578:
13577:
13576:
13572:Scientific laws
13542:
13541:
13520:Wayback Machine
13508:Project PHYSNET
13498:
13488:
13461:
13424:
13365:
13363:Related reading
13357:
13341:
13340:
13328:
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12986:
12982:
12962:
12961:
12957:
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12936:
12922:
12918:
12909:
12907:
12902:. 20 May 2019.
12890:
12889:
12885:
12879:
12868:
12862:
12861:
12857:
12848:
12846:
12833:
12832:
12828:
12819:
12817:
12810:"Coulomb's law"
12806:
12799:
12792:
12776:
12765:
12748:
12744:
12731:
12727:
12718:
12716:
12709:
12693:
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12404:
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12359:
12352:
12329:
12325:
12318:
12304:
12297:
12282:
12268:
12261:
12257:
12252:
12207:BiotâSavart law
12196:
12189:
12186:
12147:
12142:
12131:
12127:
12121:
12117:
12115:
12106:
12093:
12089:
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12077:
12073:
12072:
12071:
12057:
12053:
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12003:
11999:
11998:
11997:
11980:
11974:
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11892:
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11798:
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11716:
11715:
11698:
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11021:
11015:
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10953:
10948:
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10252:
10251:
10235:
10232:
10231:
10215:
10212:
10211:
10196:
10185:
10179:
10176:
10162:Please help by
10161:
10152:
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9180:
9169:
9164:
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9149:
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9110:
9099:
9093:
9090:
9082:help improve it
9079:
9070:
9066:
9059:
9054:
9048:
9028:BiotâSavart law
9005:
9002:
9001:
8974:
8972:
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7960:
7957:
7956:
7953:squeeze theorem
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6078:
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6028:
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5980:
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5078:
5070:
5068:
5067:is the radius,
5064:
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5017:
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5009:
5000:
4996:
4989:
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4973:
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4929:
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4910:to the article.
4901:
4896:, specifically
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2500:
2498:
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2478:
2475:
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2456:
2453:
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2442:vector addition
2434:
2410:
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2191:
2186:
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2173:
2168:
2167:
2165:
2162:
2161:
2160:
2155:, according to
2139:
2135:
2133:
2130:
2129:
2128:experienced by
2112:
2107:
2106:
2104:
2101:
2100:
2083:
2072:
2071:
2070:
2068:
2065:
2064:
2043:
2039:
2037:
2034:
2033:
2014:
2010:
2008:
2005:
2004:
2003:
1986:
1982:
1980:
1977:
1976:
1953:
1947:
1942:
1941:
1936:
1935:
1929:
1924:
1923:
1921:
1903:
1902:
1897:
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1748:
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1471:
1469:
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1437:
1436:
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1422:
1416:
1412:
1406:
1400:
1397:
1391:
1388:
1382:
1379:
1373:
1370:
1364:
1357:
1314:torsion balance
1293:
1290:
1287:
1286:
1284:
1282:
1273:
1270:
1267:
1266:
1264:
1262:
1259:Henry Cavendish
1249:
1239:torsion balance
1154:William Gilbert
1117:
1078:
1072:
1069:
1063:
1062:is a constant,
1060:
1054:
1033:
1029:
1023:
1017:
1013:
1008:
1003:
997:
993:
988:
987:
985:
979:
975:
967:
959:
957:
954:
953:
890:
861:
860:
676:
668:
667:
623:
613:
612:
568:Induction motor
538:
528:
527:
443:Current density
428:
418:
417:
408:Poynting vector
318:
316:Electrodynamics
308:
307:
303:Right-hand rule
268:Magnetic dipole
258:BiotâSavart law
248:
238:
237:
173:Electric dipole
168:Electric charge
143:
64:
58:
55:
49:
38:
28:
23:
22:
15:
12:
11:
5:
13585:
13575:
13574:
13569:
13564:
13559:
13557:Electrostatics
13554:
13540:
13539:
13534:Walter Lewin,
13529:
13523:
13510:
13497:
13496:External links
13494:
13493:
13492:
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13459:
13446:
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13405:
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13379:
13364:
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13308:
13279:
13273:. Boca Raton.
13261:
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13166:
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13123:
13105:
13077:
13059:
13031:
13007:
12980:
12955:
12934:
12916:
12883:
12877:
12855:
12839:PhysicsLab.org
12835:"Charged rods"
12826:
12797:
12790:
12763:
12742:
12725:
12707:
12687:
12670:
12663:
12645:
12639:978-0486406886
12638:
12616:Heilbron, J.L.
12607:
12590:
12577:10.1086/286445
12555:
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12523:
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12376:
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12258:
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12253:
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12250:
12248:Casimir effect
12245:
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483:Kirchhoff laws
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448:Direct current
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148:Charge density
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141:Electrostatics
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13503:Coulomb's Law
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12006:
12000:
11994:
11991:
11984:
11981:
11975:
11971:
11964:
11961:
11955:
11951:
11940:
11939:
11936:
11934:
11933:
11923:
11916:
11914:
11896:
11893:
11887:
11883:
11876:
11873:
11867:
11863:
11856:
11848:
11844:
11840:
11837:
11830:
11826:
11822:
11819:
11807:
11804:
11800:
11795:
11790:
11786:
11783:
11777:
11774:
11771:
11768:
11765:
11762:
11759:
11756:
11749:
11748:
11745:
11728:
11699:
11670:
11666:
11643:
11639:
11625:
11618:
11616:
11597:
11593:
11589:
11586:
11579:
11575:
11571:
11568:
11562:
11557:
11551:
11544:
11540:
11534:
11530:
11524:
11518:
11507:
11503:
11499:
11496:
11493:
11490:
11483:
11479:
11475:
11472:
11469:
11466:
11460:
11454:
11444:
11439:
11435:
11429:
11425:
11421:
11418:
11405:
11400:
11396:
11383:
11378:
11368:
11363:
11359:
11353:
11349:
11345:
11342:
11328:
11324:
11312:
11302:
11301:
11298:
11296:
11295:
11290:
11289:
11270:
11266:
11262:
11259:
11256:
11253:
11250:
11242:
11237:
11233:
11227:
11223:
11219:
11216:
11211:
11206:
11202:
11176:
11172:
11168:
11165:
11162:
11159:
11156:
11151:
11137:We know that
11131:
11124:
11122:
11103:
11098:
11094:
11088:
11084:
11080:
11077:
11072:
11067:
11063:
11056:
11048:
11043:
11039:
11033:
11029:
11025:
11022:
11016:
11006:
11003:
10991:
10986:
10982:
10974:
10973:
10970:
10954:
10944:
10939:
10910:
10907:
10892:
10885:
10883:
10867:
10863:
10859:
10856:
10853:
10850:
10847:
10839:
10834:
10830:
10824:
10820:
10816:
10813:
10807:
10803:
10793:
10792:
10789:
10782:
10779:Coulomb's law
10775:
10773:
10754:
10749:
10745:
10739:
10735:
10731:
10728:
10722:
10718:
10712:
10707:
10703:
10695:
10694:
10691:
10675:
10646:
10626:
10619:
10617:
10601:
10597:
10593:
10590:
10587:
10584:
10581:
10576:
10572:
10562:
10559:
10553:
10549:
10543:
10535:
10531:
10527:
10524:
10517:
10513:
10509:
10506:
10496:
10495:
10492:
10490:
10489:
10484:
10483:
10473:
10466:
10464:
10450:
10447:
10444:
10439:
10435:
10431:
10428:
10416:
10415:
10412:
10405:
10398:
10396:
10380:
10370:
10365:
10361:
10357:
10354:
10342:
10341:
10338:
10280:
10277:
10257:
10237:
10217:
10204:
10194:
10191:
10183:
10173:
10169:
10165:
10159:
10156:This section
10154:
10145:
10144:
10136:
10134:
10129:
10126:
10109:
10106:
10103:
10097:
10093:
10087:
10081:
10075:
10055:
10049:
10026:
10023:
10020:
10015:
10004:
9995:
9979:
9975:
9969:
9959:
9955:
9940:
9931:
9920:
9917:
9913:
9898:
9895:
9873:
9865:
9862:
9850:
9833:
9824:
9813:
9808:
9800:
9797:
9786:
9783:
9779:
9776:
9771:
9767:
9762:
9759:
9756:
9752:
9745:
9740:
9732:
9729:
9720:
9717:
9714:
9709:
9698:
9689:
9673:
9669:
9663:
9660:
9657:
9652:
9649:
9646:
9643:
9630:
9627:
9624:
9616:
9608:
9605:
9601:
9598:
9560:
9555:
9551:
9541:
9523:
9520:
9517:
9509:
9501:
9498:
9494:
9491:
9480:
9476:
9472:
9469:
9465:
9456:
9448:
9445:
9437:
9432:
9428:
9421:
9390:
9381:
9370:
9367:
9363:
9348:
9337:
9333:
9329:
9323:
9320:
9308:
9305:
9300:
9296:
9291:
9287:
9280:
9277:
9274:
9271:
9268:
9265:
9255:
9191:
9146:
9141:
9139:
9135:
9131:
9127:
9118:
9108:
9105:
9097:
9087:
9083:
9077:
9074:This article
9072:
9063:
9062:
9053:
9043:
9041:
9037:
9033:
9029:
9013:
9010:
9007:
8999:
8995:
8990:
8932:
8912:
8848:
8824:
8820:
8810:
8799:
8792:
8788:
8778:
8762:
8758:
8754:
8746:
8743:
8738:
8734:
8728:
8724:
8720:
8717:
8707:
8703:
8699:
8696:
8685:
8681:
8675:
8671:
8667:
8664:
8660:
8655:
8621:
8617:
8613:
8605:
8602:
8597:
8593:
8587:
8583:
8579:
8576:
8566:
8562:
8558:
8555:
8544:
8540:
8534:
8530:
8526:
8523:
8519:
8514:
8501:
8500:Lorentz force
8497:
8493:
8488:
8484:
8480:
8476:
8469:In relativity
8465:
8463:
8443:
8439:
8416:
8412:
8408:
8405:
8401:
8396:
8371:
8366:
8361:
8340:
8336:
8332:
8327:
8308:
8289:
8285:
8281:
8278:
8269:
8252:
8248:
8244:
8239:
8231:
8228:
8218:
8214:
8188:
8186:
8182:
8178:
8177:Faraday's law
8174:
8169:
8164:
8151:
8133:
8120:
8113:
8109:
8105:
8100:
8094:
8072:
8057:
8053:
8043:
8036:
8030:
8022:
8014:
7999:
7989:
7976:
7962:
7954:
7949:
7931:
7916:
7912:
7905:
7901:
7897:
7893:
7862:
7847:
7843:
7830:
7826:
7813:
7806:
7802:
7798:
7793:
7788:
7769:
7757:
7747:
7732:
7728:
7723:
7715:
7711:
7707:
7702:
7695:
7691:
7683:
7677:
7671:
7665:
7651:
7648:
7631:
7623:
7608:
7604:
7600:
7589:
7574:
7570:
7551:
7533:
7530:
7523:
7519:
7501:
7486:
7482:
7474:
7470:
7462:
7459:
7454:
7437:
7422:
7418:
7410:
7406:
7398:
7395:
7388:
7384:
7366:
7351:
7347:
7339:
7335:
7327:
7324:
7319:
7302:
7287:
7283:
7275:
7271:
7265:
7235:
7225:
7220:
7210:
7203:
7199:
7169:
7149:
7130:
7120:
7105:
7101:
7090:
7081:
7077:
7073:
7070:
7066:
7061:
7056:
7027:
7007:
6988:
6978:
6963:
6959:
6954:
6945:
6941:
6937:
6934:
6930:
6925:
6918:
6914:
6899:
6898:
6897:
6881:
6850:
6846:
6829:
6795:
6773:
6743:
6735:
6720:
6716:
6692:
6687:
6673:Now consider
6671:
6668:
6654:
6651:
6643:
6621:
6587:
6579:
6559:
6554:
6537:
6534:
6515:
6512:
6509:
6486:
6483:
6478:
6458:
6439:
6436:
6415:
6406:
6402:
6398:
6395:
6391:
6386:
6370:
6360:
6347:
6327:
6324:
6316:
6312:
6308:
6300:
6281:
6272:
6259:
6256:
6250:
6240:
6230:
6226:
6222:
6214:
6211:
6206:
6194:
6187:
6178:
6158:
6155:
6147:
6143:
6139:
6131:
6112:
6089:
6086:
6083:
6060:
6050:
6047:
6029:
6025:
6021:
6018:
6009:
5995:
5992:
5984:
5962:
5928:
5920:
5906:
5888:
5876:
5854:
5834:
5815:
5806:
5797:
5793:
5789:
5786:
5782:
5777:
5773:
5755:
5745:
5736:
5720:
5711:
5696:
5684:
5675:
5666:
5662:
5658:
5655:
5651:
5646:
5630:
5601:
5597:
5593:
5577:
5572:
5558:
5551:
5547:
5531:
5525:
5506:
5495:
5490:
5470:
5451:
5440:
5425:
5422:
5415:
5411:
5407:
5402:
5383:
5372:
5366:
5338:
5335:
5332:
5329:
5325:
5317:
5289:
5285:
5274:
5249:
5233:
5218:
5195:
5173:
5167:
5159:
5155:
5151:
5148:
5144:
5139:
5099:
5092:
5089:
5081:
5077:
5063:
5060:
5054:
5048:
5047:
5046:
5028:
5024:
5018:
5001:
4997:
4993:
4990:
4986:
4981:
4957:
4948:
4946:
4942:
4938:
4935:
4934:electrostatic
4927:
4909:
4908:summary style
4905:
4899:
4895:
4893:
4889:This article
4887:
4878:
4877:
4869:
4867:
4866:ionic bonding
4863:
4859:
4855:
4851:
4847:
4843:
4837:
4830:Atomic forces
4827:
4810:
4798:
4779:
4765:
4761:
4755:
4750:
4747:
4744:
4740:
4731:
4727:
4723:
4720:
4716:
4711:
4704:
4692:
4673:
4650:
4646:
4640:
4635:
4632:
4629:
4625:
4616:
4612:
4608:
4605:
4601:
4596:
4558:
4548:
4540:
4535:
4521:stationed at
4506:
4502:
4493:
4488:
4471:
4467:
4457:
4440:
4436:
4413:
4409:
4406:
4401:
4397:
4392:
4386:
4370:
4366:
4345:
4336:
4334:
4330:
4307:
4303:
4299:
4250:
4246:
4215:
4211:
4202:
4197:
4187:
4185:
4181:
4177:
4173:
4169:
4165:
4157:
4154:
4151:
4150:
4149:
4141:
4128:
4122:
4119:
4110:
4104:
4095:
4082:
4078:
4074:
4068:
4061:
4057:
4053:
4049:
4046:
4038:
4034:
4030:
4027:
4023:
4018:
4009:
3986:
3982:
3973:
3969:
3951:
3947:
3922:
3910:
3891:
3865:
3861:
3857:
3854:
3847:
3843:
3837:
3833:
3826:
3821:
3789:
3785:
3781:
3778:
3774:
3758:
3744:
3741:
3731:
3723:
3692:
3689:
3679:
3671:
3652:
3639:
3631:
3619:
3611:
3594:
3586:
3574:
3571:
3567:
3564:
3556:
3552:
3548:
3545:
3541:
3536:
3478:
3469:
3456:
3452:
3449:
3445:
3425:
3422:
3418:
3415:
3411:
3390:
3387:
3383:
3362:
3332:
3320:
3311:
3298:
3294:
3291:
3287:
3278:
3267:
3264:
3260:
3257:
3253:
3232:
3229:
3225:
3204:
3174:
3162:
3154:
3149:
3136:
3132:
3129:
3125:
3116:
3105:
3102:
3098:
3095:
3091:
3070:
3067:
3063:
3042:
3012:
3000:
2991:
2977:
2974:
2966:
2965:infinitesimal
2962:
2958:
2948:
2932:
2903:
2843:
2827:
2808:
2804:
2794:
2781:
2774:
2762:
2743:
2729:
2725:
2719:
2714:
2711:
2708:
2704:
2695:
2691:
2687:
2684:
2680:
2675:
2668:
2656:
2637:
2614:
2610:
2604:
2599:
2596:
2593:
2589:
2580:
2576:
2572:
2569:
2565:
2560:
2536:
2522:
2480:
2449:
2447:
2443:
2439:
2429:
2411:
2392:
2369:
2365:
2342:
2338:
2332:
2328:
2305:
2298:
2267:
2263:
2240:
2236:
2230:
2226:
2218:
2214:
2209:
2192:
2182:
2179:
2174:
2158:
2140:
2136:
2113:
2084:
2062:
2044:
2040:
2015:
2011:
1987:
1983:
1974:
1948:
1930:
1898:
1886:
1862:
1802:
1785:
1773:
1754:
1737:
1733:
1729:
1726:
1719:
1715:
1709:
1705:
1698:
1691:
1679:
1660:
1634:
1630:
1626:
1623:
1616:
1612:
1606:
1602:
1595:
1590:
1561:
1532:
1528:
1505:
1476:
1472:
1449:
1425:
1419:
1409:
1403:
1394:
1385:
1376:
1367:
1361:
1352:
1350:
1346:
1342:
1338:
1334:
1330:
1325:
1323:
1322:point charges
1319:
1315:
1311:
1307:
1302:
1260:
1255:
1252:
1247:
1240:
1235:
1231:
1229:
1226:, similar to
1225:
1221:
1217:
1212:
1210:
1209:Franz Aepinus
1206:
1202:
1198:
1194:
1190:
1185:
1183:
1182:
1177:
1176:Thomas Browne
1167:
1163:
1159:
1155:
1150:
1148:
1144:
1140:
1136:
1132:
1131:Mediterranean
1125:
1121:
1112:
1110:
1106:
1102:
1098:
1094:
1089:
1085:
1083:
1075:
1066:
1057:
1051:
1034:
1030:
1018:
1014:
998:
994:
976:
972:
964:
949:
944:
941:
937:
932:
930:
926:
922:
921:Coulomb force
918:
914:
910:
906:
902:
901:Coulomb's law
898:
887:
882:
880:
875:
873:
868:
867:
865:
864:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
678:
672:
671:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
625:
622:
617:
616:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
540:
537:
532:
531:
524:
521:
519:
516:
514:
511:
509:
506:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
478:Joule heating
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
430:
427:
422:
421:
414:
411:
409:
406:
404:
401:
399:
396:
394:
393:Lorentz force
391:
389:
386:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
320:
317:
312:
311:
304:
301:
299:
296:
294:
293:Magnetization
291:
289:
286:
284:
281:
279:
278:Magnetic flux
276:
274:
271:
269:
266:
264:
261:
259:
256:
254:
251:
250:
247:
242:
241:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
183:Electric flux
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
145:
142:
137:
136:
131:
128:
126:
123:
121:
120:Computational
118:
116:
113:
111:
108:
106:
103:
101:
98:
97:
96:
95:
91:
87:
86:
83:
80:
79:
75:
74:
71:
61:
52:
48:
47:point charges
43:
42:
37:
32:
19:
13535:
13502:
13477:
13450:
13436:
13411:
13400:
13387:
13374:
13358:
13317:
13311:
13270:
13264:
13253:. Retrieved
13241:
13229:
13196:
13176:. Retrieved
13148:
13141:
13114:
13108:
13097:. Retrieved
13087:
13080:
13069:. Retrieved
13041:
13034:
13025:
13017:
13010:
12963:
12958:
12925:
12919:
12908:. Retrieved
12895:
12886:
12864:
12858:
12847:. Retrieved
12838:
12829:
12818:. Retrieved
12780:
12756:
12745:
12735:
12728:
12717:. Retrieved
12697:
12690:
12680:
12673:
12654:
12648:
12620:
12610:
12600:
12593:
12571:(1): 75â95.
12568:
12564:
12558:
12539:
12533:
12514:
12508:
12489:
12453:
12447:
12414:
12397:
12393:
12389:
12379:
12370:
12366:
12334:
12326:
12307:
12271:
12179:
12168:
11930:
11928:
11917:
11630:
11619:
11297:), we get:
11292:
11286:
11136:
11125:
10897:
10886:
10787:
10776:
10631:
10620:
10486:
10480:
10478:
10467:
10410:
10399:
10209:
10186:
10180:October 2020
10177:
10164:spinning off
10157:
10139:Verification
10130:
10127:
9851:
9542:
9142:
9129:
9125:
9123:
9100:
9094:October 2020
9091:
9075:
8991:
8472:
8369:
8359:
8270:
8193:
8185:point charge
8180:
8167:
8160:
7977:
7950:
7652:
7649:
7552:
7252:
6830:
6672:
6669:
6348:
6274:But because
6273:
6179:
6010:
5907:
5582:
5574:
5491:
5364:
5275:
5103:
5079:
5052:
4956:point charge
4953:
4940:
4937:point charge
4931:
4890:
4839:
4491:
4490:A system of
4489:
4411:
4407:
4405:point charge
4399:
4390:
4387:
4337:
4329:point charge
4238:on a charge
4210:vector field
4207:
4161:
4147:
3970:. Using the
3764:
3653:
3491:at position
3470:
3312:
3150:
2992:
2954:
2825:
2795:
2537:
2493:at position
2450:
2435:
2282:is given by
2210:
1803:
1547:at position
1491:at position
1434:
1423:
1417:
1407:
1401:
1392:
1383:
1374:
1365:
1326:
1312:. He used a
1303:
1256:
1250:
1246:John Robison
1243:
1213:
1186:
1179:
1165:
1151:
1128:
1097:Isaac Newton
1090:
1086:
1081:
1073:
1064:
1055:
1052:
951:
946:
933:
920:
916:
900:
899:, or simply
896:
895:
638:Four-current
573:Linear motor
458:Electrolysis
338:Eddy current
298:Permeability
218:Polarization
213:Permittivity
157:
70:
59:
50:
45:between two
40:
39:
12998:|work=
12222:Gauss's law
8365:unit vector
5492:Using the "
5059:unit vector
4172:Weber force
4144:Limitations
2829:th charge,
1973:unit vector
1335:rod with a
1105:Gauss's law
608:Transformer
438:Capacitance
363:Faraday law
158:Coulomb law
100:Electricity
13546:Categories
13469:2007010418
13336:1176566442
13289:1028553174
13255:2020-09-24
13178:2022-10-09
13099:2022-10-09
13071:2022-10-09
12910:2019-05-20
12849:2007-11-06
12820:2007-11-30
12719:2009-10-17
12343:. p.
12255:References
11285:Dividing (
10479:Dividing (
10168:relocating
9134:scattering
9050:See also:
8496:four force
8206:, we have
6073:such that
4892:duplicates
4834:See also:
1349:instrument
1333:insulating
1237:Coulomb's
1166:electricus
675:Scientists
523:Waveguides
503:Resistance
473:Inductance
253:AmpĂšre law
13434:(1979) .
13344:cite book
13297:cite book
13215:cite book
13000:ignored (
12990:cite book
12944:cite book
12701:. Wiley.
12585:121067746
12433:cite book
12290:739118459
12140:≈
12113:⇒
12066:≈
12039:⇒
11992:≈
11985:ℓ
11965:ℓ
11897:ℓ
11877:ℓ
11857:≈
11845:θ
11841:
11827:θ
11823:
11814:⇒
11808:ℓ
11791:ℓ
11775:θ
11772:
11766:≈
11763:θ
11760:
11667:θ
11640:θ
11594:θ
11590:
11576:θ
11572:
11516:⇒
11504:θ
11500:
11480:θ
11476:
11426:ε
11422:π
11350:ε
11346:π
11267:θ
11263:
11224:ε
11220:π
11173:θ
11169:
11085:ε
11081:π
11030:ε
11026:π
10864:θ
10860:
10821:ε
10817:π
10736:ε
10732:π
10598:θ
10594:
10569:⇒
10532:θ
10528:
10514:θ
10510:
10436:θ
10432:
10362:θ
10358:
10107:π
10053:→
10050:ε
10027:ε
10021:−
9996:−
9932:−
9918:−
9896:∫
9825:−
9814:δ
9801:π
9768:−
9746:δ
9721:ε
9715:−
9690:−
9661:−
9634:⟩
9595:⟨
9561:≫
9527:⟩
9488:⟨
9449:π
9422:∫
9382:−
9368:−
9330:∫
9321:−
9297:−
9281:δ
9278:π
9266:−
9260:⟩
9242:→
9239:⟩
9196:⟩
9178:→
9175:⟩
9011:≪
9008:β
8933:θ
8913:β
8811:×
8779:×
8747:θ
8744:
8725:β
8721:−
8704:β
8700:−
8672:ϵ
8668:π
8606:θ
8603:
8584:β
8580:−
8563:β
8559:−
8531:ϵ
8527:π
8434:^
8413:ε
8409:π
8337:ε
8309:⋅
8303:^
8282:π
8249:ε
8229:⋅
8215:∮
8121:ρ
8110:ε
8089:Φ
8034:→
7990:⋅
7986:∇
7963:ρ
7906:∈
7814:ρ
7803:ε
7758:ρ
7724:∫
7712:ε
7692:ε
7660:Φ
7635:∅
7601:∩
7567:∖
7564:Ω
7531:⋅
7479:∂
7475:∮
7460:⋅
7415:∂
7411:∮
7396:⋅
7344:∂
7340:∮
7325:⋅
7280:∂
7276:∮
7260:Φ
7098:∖
7095:Ω
7091:∫
7078:ε
7074:π
6955:∫
6942:ε
6938:π
6816:Ω
6747:Ω
6744:⊆
6696:Ω
6693:∈
6622:⋅
6611:∇
6602:and then
6580:≠
6563:Ω
6560:∈
6547:∀
6535:∈
6527:∀
6523:⟹
6519:∅
6510:∩
6507:Ω
6437:⋅
6426:∇
6420:Ω
6416:∫
6403:ε
6399:π
6361:⋅
6357:∇
6331:Ω
6328:×
6309:∈
6241:⋅
6237:∇
6227:∫
6212:⋅
6192:∂
6188:∮
6162:Ω
6159:×
6140:∈
6093:∅
6084:∩
6081:Ω
6058:∂
6046:piecewise
6044:having a
6022:⊆
5963:⋅
5952:∇
5921:≠
5877:ρ
5811:Ω
5807:∫
5794:ε
5790:π
5778:≡
5737:−
5712:−
5685:ρ
5680:Ω
5676:∫
5663:ε
5659:π
5594:⊆
5591:Ω
5548:ε
5532:ρ
5507:⋅
5504:∇
5452:−
5441:δ
5426:ρ
5423:∫
5412:ε
5384:⋅
5381:∇
5339:δ
5336:π
5286:⋅
5283:∇
5219:−
5196:−
5174:ρ
5168:∫
5156:ε
5152:π
4998:ε
4994:π
4858:molecules
4854:electrons
4741:∑
4728:ε
4724:π
4667:^
4626:∑
4613:ε
4609:π
4549:−
4120:−
4111:⋅
4096:⋅
4075:×
4035:ε
4031:π
3983:ε
3948:ε
3885:^
3862:ε
3858:π
3786:ε
3782:π
3724:−
3672:−
3612:−
3587:−
3565:∫
3553:ε
3549:π
3426:ρ
3321:ρ
3268:σ
3163:σ
3153:capacitor
3130:ℓ
3106:λ
3068:ℓ
3001:λ
2897:^
2705:∑
2692:ε
2688:π
2631:^
2590:∑
2577:ε
2573:π
2405:^
2393:−
2299:^
2183:−
2078:^
2041:ε
1880:^
1836:−
1734:ε
1730:π
1654:^
1631:ε
1627:π
1218:spheres,
1205:capacitor
1184:of 1646.
1162:Neo-Latin
1158:lodestone
1091:Being an
948:distance.
831:Steinmetz
761:Kirchhoff
746:Jefimenko
741:Hopkinson
726:Helmholtz
721:Heaviside
583:Permeance
468:Impedance
208:Insulator
203:Gauss law
153:Conductor
130:Phenomena
125:Textbooks
105:Magnetism
18:Coulombic
13516:Archived
13246:Archived
13172:Archived
13133:40251748
13093:Archived
13065:Archived
12904:Archived
12843:Archived
12814:Archived
12713:Archived
12618:(1979).
12456:: 3â20.
12184:See also
10005:′
9941:′
9834:′
9780:′
9699:′
9602:′
9495:′
9391:′
9309:′
9256:′
9192:′
7902:′
7831:′
7789:′
7770:′
7170:′
7150:′
7028:′
7008:′
6644:′
6588:′
6555:′
6479:′
6459:′
6301:′
6132:′
5985:′
5929:′
5889:′
5855:′
5835:′
5774:′
5751:‖
5746:′
5728:‖
5721:′
5697:′
4180:Einstein
4168:magnetic
3732:′
3680:′
3620:′
3595:′
3575:′
3453:′
3437:′
3419:′
3391:′
3363:′
3333:′
3295:′
3279:′
3261:′
3233:′
3205:′
3175:′
3155:) where
3133:′
3117:′
3099:′
3071:′
3043:′
3013:′
2961:integral
2063:. Here,
1171:ጀλΔÎșÏÏÎżÎœ
1147:friction
856:Wiechert
811:Poynting
701:Einstein
548:DC motor
543:AC motor
378:Lenz law
163:Electret
12626:460â462
9080:Please
8494:of the
8194:Taking
6788:having
5369:is the
5086:is the
4902:Please
3966:is the
2217:product
1859:is the
1399:. When
1297:
1285:
1277:
1265:
1193:gravity
1115:History
940:charges
909:physics
841:Thomson
816:Ritchie
806:Poisson
791:Neumann
786:Maxwell
781:Lorentz
776:Liénard
706:Faraday
691:Coulomb
518:Voltage
493:Ohm law
115:History
13484:
13467:
13457:
13420:
13334:
13324:
13287:
13277:
13203:
13164:
13131:
13121:
13057:
12978:
12932:
12875:
12788:
12705:
12661:
12636:
12583:
12546:
12521:
12496:
12468:
12421:
12314:
12288:
12278:
11291:) by (
10485:) by (
9143:Under
8841:where
8357:where
7909:
6708:, and
6572:
6569:
6566:
6544:
6541:
5362:where
5269:where
5098:origin
5073:|
5069:|
5045:where
4088:
3972:CODATA
3376:, and
3218:, and
3056:, and
2796:where
2451:Force
1919:
1892:
1804:where
1429:< 0
1413:> 0
1207:, and
1053:Here,
826:Singer
821:Savart
801:Ărsted
766:Larmor
756:Kelvin
711:Fizeau
681:AmpĂšre
603:Stator
110:Optics
67:other.
13567:Force
13249:(PDF)
13238:(PDF)
12869:(PDF)
12581:S2CID
12392:[
12369:[
11191:and:
8363:is a
8171:(see
7885:with
7042:,
5943:that
4846:force
4842:atoms
4050:8.987
2159:, is
1337:metal
1189:force
1164:word
1135:amber
936:force
851:Weber
846:Volta
836:Tesla
751:Joule
736:Hertz
731:Henry
716:Gauss
598:Rotor
36:force
13482:ISBN
13465:LCCN
13455:ISBN
13444:â27.
13418:ISBN
13350:link
13332:OCLC
13322:ISBN
13303:link
13285:OCLC
13275:ISBN
13221:link
13201:ISBN
13162:ISBN
13129:OCLC
13119:ISBN
13055:ISBN
13002:help
12976:ISBN
12950:link
12930:ISBN
12900:NIST
12873:ISBN
12786:ISBN
12703:ISBN
12659:ISBN
12634:ISBN
12628:and
12544:ISBN
12519:ISBN
12494:ISBN
12466:ISBN
12439:link
12419:ISBN
12312:ISBN
12286:OCLC
12276:ISBN
11714:and
11658:and
10945:<
10788:so:
10632:Let
10411:and
9211:is:
9124:The
8967:and
8175:and
8163:curl
7184:and
6867:and
6831:Let
5583:Let
4958:is:
4862:ions
4178:and
2880:and
2436:The
2213:sign
2059:the
2032:and
1341:silk
1283:2 â
1263:2 +
1199:and
1071:and
771:Lenz
696:Davy
686:Biot
57:and
13506:on
13154:doi
13047:doi
13022:or
12968:doi
12630:464
12573:doi
12458:doi
11838:tan
11820:tan
11769:sin
11757:tan
11587:tan
11569:tan
11497:tan
11473:tan
11260:tan
11166:tan
10857:tan
10591:tan
10525:cos
10507:sin
10491:):
10429:cos
10355:sin
10166:or
9084:to
8735:sin
8594:sin
8181:can
8165:of
8027:lim
5367:(r)
4941:can
4182:'s
4058:792
4054:551
2384:is
2002:to
1178:'s
919:or
907:of
905:law
796:Ohm
13548::
13463:.
13442:23
13399:.
13386:.
13373:.
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13342:{{
13330:.
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13295:{{
13283:.
13244:.
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12898:.
12894:.
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12837:.
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12766:^
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12711:.
12579:.
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12431:{{
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12262:^
8989:.
8360:rÌ
6667:)
6346:,
6008:.
4335:.
4079:10
4069:14
3911:12
3892:12
2947:.
2412:12
2306:12
2085:12
1971:a
1949:12
1931:12
1887:12
1818:12
1774:12
1755:12
1680:12
1661:12
1299:th
1294:50
1279:th
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1254:.
13490:.
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12938:.
12913:.
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12823:.
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12722:.
12667:.
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12587:.
12575::
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12527:.
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12427:.
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12320:.
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9970:=
9966:r
9960:3
9956:d
9949:r
9945:)
9937:p
9928:p
9924:(
9921:i
9914:e
9910:)
9906:r
9902:(
9899:V
9874:2
9870:)
9866:m
9863:2
9860:(
9838:)
9830:p
9821:p
9817:(
9809:4
9805:)
9798:2
9795:(
9792:)
9787:k
9784:,
9777:p
9772:E
9763:k
9760:,
9757:p
9753:E
9749:(
9741:2
9737:)
9733:m
9730:2
9727:(
9718:i
9710:2
9705:|
9695:p
9686:p
9681:|
9674:2
9670:e
9664:i
9658:=
9653:n
9650:n
9647:o
9644:c
9639:|
9631:k
9628:,
9625:p
9621:|
9617:S
9613:|
9609:k
9606:,
9599:p
9575:|
9570:p
9565:|
9556:0
9552:m
9524:k
9521:,
9518:p
9514:|
9510:S
9506:|
9502:k
9499:,
9492:p
9481:0
9477:r
9473:k
9470:i
9466:e
9457:3
9453:)
9446:2
9443:(
9438:k
9433:3
9429:d
9399:r
9395:)
9387:p
9378:p
9374:(
9371:i
9364:e
9360:)
9356:r
9352:(
9349:V
9344:r
9338:3
9334:d
9327:)
9324:i
9318:(
9315:)
9306:p
9301:E
9292:p
9288:E
9284:(
9275:2
9272:=
9269:1
9263:)
9252:p
9246:|
9235:p
9230:|
9226:(
9221:A
9199:)
9188:p
9182:|
9171:p
9166:|
9162:(
9157:A
9130:E
9107:)
9101:(
9096:)
9092:(
9078:.
9014:1
8976:v
8954:r
8892:v
8870:r
8849:q
8825:2
8821:c
8815:E
8807:v
8800:=
8793:2
8789:c
8783:r
8775:v
8763:2
8759:/
8755:3
8751:)
8739:2
8729:2
8718:1
8715:(
8708:2
8697:1
8686:3
8682:r
8676:0
8665:4
8661:q
8656:=
8652:B
8631:r
8622:2
8618:/
8614:3
8610:)
8598:2
8588:2
8577:1
8574:(
8567:2
8556:1
8545:3
8541:r
8535:0
8524:4
8520:q
8515:=
8511:E
8444:2
8440:r
8430:r
8417:0
8406:4
8402:Q
8397:=
8394:)
8390:r
8386:(
8382:E
8370:E
8341:0
8333:Q
8328:=
8325:)
8321:r
8317:(
8313:E
8299:r
8290:2
8286:r
8279:4
8253:0
8245:Q
8240:=
8236:A
8232:d
8225:E
8219:S
8204:Q
8200:r
8196:S
8168:E
8139:)
8134:0
8129:r
8124:(
8114:0
8106:1
8101:=
8098:)
8095:R
8092:(
8082:|
8078:)
8073:0
8068:r
8063:(
8058:R
8054:B
8049:|
8044:1
8037:0
8031:R
8023:=
8020:)
8015:0
8010:r
8005:(
8000:0
7995:E
7937:)
7932:0
7927:r
7922:(
7917:R
7913:B
7898:c
7894:r
7872:|
7868:)
7863:0
7858:r
7853:(
7848:R
7844:B
7839:|
7835:)
7827:c
7822:r
7817:(
7807:0
7799:1
7794:=
7785:r
7779:d
7774:)
7766:r
7761:(
7753:)
7748:0
7743:r
7738:(
7733:R
7729:B
7716:0
7708:1
7703:=
7696:0
7687:)
7684:R
7681:(
7678:Q
7672:=
7669:)
7666:R
7663:(
7632:=
7629:)
7624:0
7619:r
7614:(
7609:R
7605:B
7598:)
7595:)
7590:0
7585:r
7580:(
7575:R
7571:B
7561:(
7538:S
7534:d
7524:R
7520:B
7514:E
7507:)
7502:0
7497:r
7492:(
7487:R
7483:B
7471:=
7467:S
7463:d
7455:C
7450:E
7443:)
7438:0
7433:r
7428:(
7423:R
7419:B
7407:+
7403:S
7399:d
7389:R
7385:B
7379:E
7372:)
7367:0
7362:r
7357:(
7352:R
7348:B
7336:=
7332:S
7328:d
7320:0
7315:E
7308:)
7303:0
7298:r
7293:(
7288:R
7284:B
7272:=
7269:)
7266:R
7263:(
7236:0
7231:E
7226:=
7221:C
7216:E
7211:+
7204:R
7200:B
7194:E
7166:r
7160:d
7155:)
7146:r
7141:,
7138:r
7134:(
7131:e
7126:)
7121:0
7116:r
7111:(
7106:R
7102:B
7082:0
7071:4
7067:1
7062:=
7057:C
7052:E
7024:r
7018:d
7013:)
7004:r
6999:,
6996:r
6992:(
6989:e
6984:)
6979:0
6974:r
6969:(
6964:R
6960:B
6946:0
6935:4
6931:1
6926:=
6919:R
6915:B
6909:E
6882:C
6877:E
6851:R
6847:B
6841:E
6796:R
6774:0
6769:r
6741:)
6736:0
6731:r
6726:(
6721:R
6717:B
6688:0
6683:r
6655:0
6652:=
6649:)
6641:r
6637:,
6634:r
6630:(
6626:e
6616:r
6585:r
6576:r
6552:r
6538:V
6531:r
6516:=
6513:V
6487:0
6484:=
6475:r
6469:d
6464:)
6455:r
6450:,
6447:r
6443:(
6440:e
6431:r
6407:0
6396:4
6392:1
6387:=
6384:)
6380:r
6376:(
6371:0
6366:E
6334:)
6325:V
6322:(
6317:1
6313:C
6306:)
6297:r
6292:,
6289:r
6285:(
6282:e
6260:V
6257:d
6251:0
6246:E
6231:V
6223:=
6219:S
6215:d
6207:0
6202:E
6195:V
6165:)
6156:V
6153:(
6148:1
6144:C
6137:)
6128:r
6123:,
6120:r
6116:(
6113:e
6090:=
6087:V
6061:V
6030:3
6026:R
6019:V
5996:0
5993:=
5990:)
5982:r
5978:,
5975:r
5971:(
5967:e
5957:r
5926:r
5917:r
5893:)
5885:r
5880:(
5851:r
5845:d
5840:)
5831:r
5826:,
5823:r
5819:(
5816:e
5798:0
5787:4
5783:1
5770:r
5764:d
5756:3
5742:r
5733:r
5717:r
5708:r
5701:)
5693:r
5688:(
5667:0
5656:4
5652:1
5647:=
5644:)
5640:r
5636:(
5631:0
5626:E
5602:3
5598:R
5559:,
5552:0
5543:)
5539:r
5535:(
5526:=
5523:)
5519:r
5515:(
5511:E
5477:s
5471:3
5466:d
5460:)
5456:s
5448:r
5444:(
5437:)
5433:s
5429:(
5416:0
5408:1
5403:=
5400:)
5396:r
5392:(
5388:E
5365:ÎŽ
5350:)
5346:r
5342:(
5333:4
5330:=
5326:)
5318:3
5313:|
5307:r
5302:|
5296:r
5290:(
5271:Ï
5256:s
5250:3
5245:d
5234:3
5229:|
5223:s
5215:r
5210:|
5204:)
5200:s
5192:r
5188:(
5185:)
5181:s
5177:(
5160:0
5149:4
5145:1
5140:=
5137:)
5133:r
5129:(
5125:E
5114:s
5110:r
5106:r
5100:.
5094:q
5090:,
5083:0
5080:Δ
5075:,
5071:r
5065:r
5061:,
5053:r
5050:e
5029:2
5025:r
5019:r
5014:e
5002:0
4991:4
4987:q
4982:=
4979:)
4975:r
4971:(
4967:E
4928:.
4900:.
4811:3
4805:|
4799:i
4794:r
4788:|
4780:i
4775:r
4766:i
4762:q
4756:n
4751:1
4748:=
4745:i
4732:0
4721:4
4717:1
4712:=
4705:2
4699:|
4693:i
4688:r
4682:|
4674:i
4663:r
4651:i
4647:q
4641:n
4636:1
4633:=
4630:i
4617:0
4606:4
4602:1
4597:=
4594:)
4590:r
4586:(
4582:E
4559:i
4554:r
4545:r
4541:=
4536:i
4531:r
4507:i
4503:q
4492:n
4472:2
4468:r
4462:|
4458:q
4454:|
4445:e
4441:k
4437:=
4433:|
4428:E
4423:|
4412:r
4408:Q
4400:E
4391:E
4371:t
4367:q
4346:q
4314:E
4308:t
4304:q
4300:=
4296:F
4274:E
4251:t
4247:q
4225:F
4129:.
4123:2
4116:C
4105:2
4101:m
4092:N
4083:9
4072:)
4066:(
4062:3
4047:=
4039:0
4028:4
4024:1
4019:=
4014:e
4010:k
3987:0
3952:0
3923:2
3917:|
3906:r
3900:|
3881:r
3866:0
3855:4
3848:2
3844:q
3838:1
3834:q
3827:=
3822:1
3817:F
3790:0
3779:4
3775:1
3745:0
3742:=
3738:|
3729:r
3720:r
3715:|
3693:0
3690:=
3686:|
3677:r
3668:r
3663:|
3640:.
3632:3
3627:|
3617:r
3608:r
3603:|
3592:r
3583:r
3572:q
3568:d
3557:0
3546:4
3542:q
3537:=
3534:)
3530:r
3526:(
3522:F
3500:r
3479:q
3457:.
3450:V
3446:d
3442:)
3434:r
3429:(
3423:=
3416:q
3412:d
3388:V
3384:d
3359:r
3337:)
3329:r
3324:(
3299:.
3292:A
3288:d
3284:)
3276:r
3271:(
3265:=
3258:q
3254:d
3230:A
3226:d
3201:r
3179:)
3171:r
3166:(
3137:.
3126:d
3122:)
3114:r
3109:(
3103:=
3096:q
3092:d
3064:d
3039:r
3017:)
3009:r
3004:(
2978:q
2975:d
2933:i
2928:r
2904:i
2893:r
2867:r
2844:i
2839:r
2826:i
2809:i
2805:q
2782:,
2775:3
2769:|
2763:i
2758:r
2752:|
2744:i
2739:r
2730:i
2726:q
2720:n
2715:1
2712:=
2709:i
2696:0
2685:4
2681:q
2676:=
2669:2
2663:|
2657:i
2652:r
2646:|
2638:i
2627:r
2615:i
2611:q
2605:n
2600:1
2597:=
2594:i
2581:0
2570:4
2566:q
2561:=
2558:)
2554:r
2550:(
2546:F
2523:n
2502:r
2481:q
2460:F
2426:;
2401:r
2370:1
2366:q
2343:2
2339:q
2333:1
2329:q
2295:r
2268:1
2264:q
2241:2
2237:q
2231:1
2227:q
2207:.
2193:1
2188:F
2180:=
2175:2
2170:F
2141:2
2137:q
2114:2
2109:F
2075:r
2045:0
2030:,
2016:1
2012:q
1988:2
1984:q
1955:|
1944:r
1938:|
1926:r
1911:f
1908:e
1905:d
1899:=
1876:r
1844:2
1840:r
1831:1
1827:r
1823:=
1814:r
1786:3
1780:|
1769:r
1763:|
1750:r
1738:0
1727:4
1720:2
1716:q
1710:1
1706:q
1699:=
1692:2
1686:|
1675:r
1669:|
1650:r
1635:0
1624:4
1617:2
1613:q
1607:1
1603:q
1596:=
1591:1
1586:F
1562:2
1557:r
1533:2
1529:q
1506:1
1501:r
1477:1
1473:q
1450:1
1445:F
1427:2
1424:q
1421:1
1418:q
1411:2
1408:q
1405:1
1402:q
1396:2
1393:q
1387:2
1384:F
1378:1
1375:q
1369:1
1366:F
1291:/
1288:1
1271:/
1268:1
1251:x
1082:r
1077:2
1074:q
1068:1
1065:q
1059:e
1056:k
1035:2
1031:r
1025:|
1019:2
1015:q
1010:|
1005:|
999:1
995:q
990:|
981:e
977:k
973:=
969:|
965:F
961:|
885:e
878:t
871:v
63:2
60:q
54:1
51:q
41:F
20:)
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